{"id":9941,"date":"2026-07-03T17:39:42","date_gmt":"2026-07-03T17:39:42","guid":{"rendered":"https:\/\/kapdec.com\/help\/?p=9941"},"modified":"2026-07-03T17:39:42","modified_gmt":"2026-07-03T17:39:42","slug":"classification-of-groups","status":"publish","type":"post","link":"https:\/\/kapdec.com\/help\/classification-of-groups\/","title":{"rendered":"Classification Of Groups"},"content":{"rendered":"
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KAPDEC® | Elite STEM Learning Platform | https:\/\/kapdec.com<\/a><\/p>\n

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Unit: <\/strong>Analogy & Classifications<\/strong><\/h2>\n

Chapter: <\/strong>Classification of Groups<\/strong><\/h3>\n

Reference: – Introduction to Classification, Number Classification, Letter\/Alphabet Classification, Word Classification, General Knowledge Classification, Coding-Based Classification, Visual Classification, Mixed Classification Problems, Odd-One-Out (Outlier Detection), Venn Diagram-Based Classification, Family Relationship Classification, Direction-Based Classification<\/em><\/p>\n

\u00a0<\/p>\n

After studying this chapter, you should be able to understand:<\/strong><\/p>\n

    \n
  • Introduction to Classification<\/li>\n
  • Number Classification or Letter\/Alphabet Classification<\/li>\n
  • Word & General Classification<\/li>\n
  • Coding Based & Conclusion classifications<\/li>\n<\/ul>\n

    Introduction to Classification<\/strong><\/p>\n

    Definition<\/u><\/strong><\/p>\n

    Classification is a fundamental reasoning process in which objects, numbers, letters, words, or even ideas are grouped based on a common property or relationship.
    \nThe purpose is to recognize patterns and identify similarities or differences among given items.<\/p>\n

    When we classify, we essentially ask:<\/p>\n

    \u201cWhat is the common feature among all these items?\u201d<\/p>\n

    Once we identify the feature, we can determine which element does not belong or predict which element could complete the group.<\/p>\n

    Importance of Classification<\/u><\/strong><\/p>\n

      \n
    • Helps in organizing knowledge and making sense of information.<\/li>\n
    • Improves logical reasoning and analytical skills.<\/li>\n
    • Used in exams (logical reasoning section), data science, biology, library science, and everyday decision-making.<\/li>\n
    • Allows us to see relationships and make generalizations.<\/li>\n<\/ul>\n

      Example<\/strong><\/p>\n

      Group:<\/strong> {Apple, Banana, Mango, Orange}
      \nCommon Property:<\/strong> All are fruits.
      \nSo, if \u201cCarrot\u201d was given as an option, we could say it does not belong (since carrot is a vegetable).<\/p>\n

      Subtopics<\/u><\/strong><\/p>\n

      1. Concept of Homogeneity<\/strong><\/p>\n

      Homogeneity means similarity in nature.
      \nIn classification problems, all members of a group must share a homogeneous property.<\/p>\n

        \n
      • If the group is {2, 4, 6, 8}, the homogeneity is that all are even numbers.<\/li>\n
      • If the group is {Rose, Lily, Jasmine, Lotus}, the homogeneity is that all are flowers.<\/li>\n<\/ul>\n

        Key Points:<\/strong><\/p>\n

          \n
        • Homogeneity ensures logical consistency of the group.<\/li>\n
        • If one member does not share the common property, it is considered an odd one out.<\/li>\n<\/ul>\n

          2. Finding the Group Basis (Property)<\/strong><\/p>\n

          The group basis is the common property that connects all members of a group.
          \nFinding this property is the central step in solving classification problems.<\/p>\n

          Steps to Identify Group Basis:<\/strong><\/p>\n

            \n
          1. Observe<\/strong> all items carefully.<\/li>\n
          2. Compare<\/strong> them to see what they have in common.<\/li>\n
          3. Check category types<\/strong> (object, number, letter, word, shape).<\/li>\n
          4. Identify property<\/strong> (e.g., type, function, color, origin, use).<\/li>\n<\/ol>\n

            Example 1 \u2013 Words:<\/strong>
            \nGroup: {Mercury, Venus, Earth, Mars}
            \nCommon Property: All are planets in the solar system.<\/p>\n

            Example 2 \u2013 Numbers:<\/strong>
            \nGroup: {3, 5, 7, 11}
            \nCommon Property: All are prime numbers.<\/p>\n

            Example 3 \u2013 Letters:<\/strong>
            \nGroup: {A, E, I, O, U}
            \nCommon Property: All are vowels.<\/p>\n

            \"\"<\/p>\n

            Source: Kapdec.com<\/p>\n<\/div>\n

            Number Classification<\/strong><\/p>\n

            Definition<\/u><\/strong><\/p>\n

            Number Classification is the process of grouping numbers based on their mathematical properties.
            \nThis is widely used in reasoning tests, mathematics, and problem-solving to quickly identify patterns among numbers.<\/p>\n

            Importance of Number Classification<\/strong><\/p>\n

              \n
            • Helps in quick pattern recognition.<\/li>\n
            • Strengthens mathematical reasoning skills.<\/li>\n
            • Useful in competitive exams (reasoning section).<\/li>\n
            • Improves ability to solve series, sequences, and puzzle problems.<\/li>\n<\/ul>\n

              Examples<\/strong><\/p>\n

                \n
              • Group 1:<\/strong> {2, 4, 6, 8} \u2192 All are even numbers.<\/li>\n
              • Group 2:<\/strong> {3, 5, 7, 11} \u2192 All are prime numbers.<\/li>\n<\/ul>\n

                Subtopics<\/u><\/strong><\/p>\n

                1. Even and Odd Numbers<\/strong><\/p>\n

                Numbers are classified based on divisibility by 2.<\/p>\n

                  \n
                • Even Numbers<\/strong> \u2192 Divisible by 2, leave no remainder.
                  \n\tExamples: 2, 4, 6, 8, 10…<\/li>\n
                • Odd Numbers<\/strong> \u2192 Not divisible by 2, leave a remainder of 1.
                  \n\tExamples: 1, 3, 5, 7, 9…<\/li>\n<\/ul>\n

                  Quick Tip:<\/strong>
                  \nEven number = 2 \u00d7 (any integer)
                  \nOdd number = 2 \u00d7 (any integer) + 1<\/p>\n

                  2. Prime and Composite Numbers<\/strong><\/p>\n

                  Numbers classified based on number of factors.<\/p>\n

                    \n
                  • Prime Numbers<\/strong> \u2192 Have exactly two factors (1 and itself).
                    \n\tExamples: 2, 3, 5, 7, 11, 13…
                    \n\t(Note: 2 is the only even prime number.)<\/li>\n
                  • Composite Numbers<\/strong> \u2192 Have more than two factors.
                    \n\tExamples: 4 (1, 2, 4), 6 (1, 2, 3, 6), 9 (1, 3, 9).<\/li>\n<\/ul>\n

                    Special Note:<\/u><\/strong>
                    \n1 is neither prime nor composite (it has only one factor).<\/p>\n

                    3. Perfect Squares and Cubes<\/strong><\/p>\n

                    Numbers classified based on whether they are squares\/cubes of integers.<\/p>\n

                      \n
                    • Perfect Squares<\/strong> \u2192 Numbers obtained by squaring an integer.
                      \n\tExamples: 1 (1\u00b2), 4 (2\u00b2), 9 (3\u00b2), 16 (4\u00b2)…<\/li>\n
                    • Perfect Cubes<\/strong> \u2192 Numbers obtained by cubing an integer.
                      \n\tExamples: 1 (1\u00b3), 8 (2\u00b3), 27 (3\u00b3), 64 (4\u00b3)…<\/li>\n<\/ul>\n

                      These are useful for recognizing series and solving puzzles.<\/p>\n

                      4. Multiples and Factors<\/strong><\/p>\n

                      Numbers classified by relationship with other numbers.<\/p>\n

                        \n
                      • Multiples<\/strong> \u2192 Numbers obtained by multiplying a number by integers.
                        \n\tExample: Multiples of 3 \u2192 3, 6, 9, 12, 15…<\/li>\n
                      • Factors<\/strong> \u2192 Numbers that divide a given number completely.
                        \n\tExample: Factors of 12 \u2192 1, 2, 3, 4, 6, 12.<\/li>\n<\/ul>\n

                        Quick Rule:<\/strong><\/p>\n

                          \n
                        • Factors are limited<\/strong> (finite).<\/li>\n
                        • Multiples are infinite<\/strong>.<\/li>\n<\/ul>\n

                          5. Arithmetic and Geometric Progressions<\/strong><\/p>\n

                          Numbers can be grouped based on their arrangement in a sequence.<\/p>\n

                            \n
                          • Arithmetic Progression (AP)<\/strong>
                            \n\tA sequence where the difference between consecutive terms is constant.
                            \n\tExample: 2, 4, 6, 8, 10 (Common difference = 2)<\/li>\n
                          • Geometric Progression (GP)<\/strong>
                            \n\tA sequence where the ratio between consecutive terms is constant.
                            \n\tExample: 3, 6, 12, 24, 48 (Common ratio = 2)<\/li>\n<\/ul>\n

                            Recognizing AP and GP helps solve series problems quickly.<\/p>\n

                            Letter \/ Alphabet Classification<\/strong><\/p>\n

                            Definition<\/u><\/strong><\/p>\n

                            Letter or Alphabet Classification is the process of grouping letters of the alphabet based on their position, sequence, or specific pattern.<\/p>\n

                            This is a common topic in reasoning tests, where the task is to find the common relationship among given letters and sometimes identify the letter that does not belong.<\/p>\n

                            Importance of Letter Classification<\/strong><\/p>\n

                              \n
                            • Improves alphabet familiarity and observation skills.<\/li>\n
                            • Builds logical reasoning and pattern recognition.<\/li>\n
                            • Useful for verbal reasoning, series completion, and coding-decoding problems.<\/li>\n<\/ul>\n

                              Examples<\/strong><\/p>\n

                                \n
                              • Group 1:<\/strong> {A, E, I, O, U} \u2192 All are vowels.<\/li>\n
                              • Group 2:<\/strong> {B, D, F, H} \u2192 Alternate letters starting from B (B +2 = D, D +2 = F, etc.).<\/li>\n<\/ul>\n

                                Subtopics<\/u><\/strong><\/p>\n

                                1. Vowels and Consonants<\/strong><\/p>\n

                                The most basic way to classify letters is by separating vowels and consonants.<\/p>\n

                                  \n
                                • Vowels:<\/strong> A, E, I, O, U<\/li>\n
                                • Consonants:<\/strong> All other letters (B, C, D, F, G, … Z)<\/li>\n<\/ul>\n

                                  Example:<\/strong>
                                  \nGroup: {A, E, I, O, U} \u2192 All are vowels.
                                  \nOdd one out in {A, E, I, K, O} \u2192 K<\/strong> (because it is a consonant).<\/p>\n

                                  2. Position-Based Classification<\/strong><\/p>\n

                                  Letters are classified based on their position in the alphabet.<\/p>\n

                                  (a) Consecutive Letters<\/strong><\/p>\n

                                    \n
                                  • Letters that come one after the other in the alphabet.
                                    \n\tExample: {L, M, N, O} \u2192 Consecutive letters (12th, 13th, 14th, 15th positions).<\/li>\n<\/ul>\n

                                    (b) Even \/ Odd Position Letters<\/strong><\/p>\n

                                      \n
                                    • Even-position letters:<\/strong> B (2), D (4), F (6)…<\/li>\n
                                    • Odd-position letters:<\/strong> A (1), C (3), E (5)…<\/li>\n<\/ul>\n

                                      Example: {B, D, F, H} \u2192 All letters at even positions in the alphabet.<\/p>\n

                                      3. Reverse Alphabet Order<\/strong><\/p>\n

                                      Letters classified based on their position when the alphabet is written backwards (Z to A).<\/p>\n

                                      Example:
                                      \nReverse Alphabet: Z(1), Y(2), X(3)…
                                      \nGroup: {Z, Y, X, W} \u2192 First four letters in reverse order.<\/p>\n

                                      Odd one out in {M, N, O, L} \u2192 L<\/strong> (not in correct consecutive reverse order with others).<\/p>\n

                                      4. Skip-Sequence Patterns<\/strong><\/p>\n

                                      Letters follow a fixed skipping pattern in the alphabet.<\/p>\n

                                      Examples:<\/p>\n

                                        \n
                                      • Skip-1 pattern: A, C, E, G, I (each letter skips one letter in between)<\/li>\n
                                      • Skip-2 pattern: A, D, G, J (each letter skips two letters in between)<\/li>\n<\/ul>\n

                                        Example Question:<\/strong>
                                        \nGroup: {A, C, E, G} \u2192 All letters have a gap of one between them.
                                        \nOdd one out in {A, C, E, H} \u2192 H<\/strong> (does not follow skip-1 pattern).<\/p>\n

                                        Word Classification<\/strong><\/p>\n

                                        Definition<\/u><\/strong><\/p>\n

                                        Word Classification is the process of grouping words based on their meaning, category, or functional relationship.
                                        \nIt focuses on the semantic connection between words rather than just their spelling or sound.<\/p>\n

                                        The goal is to identify what all given words have in common and use that common property to solve classification or odd-one-out problems.<\/p>\n

                                        Importance of Word Classification<\/strong><\/p>\n

                                          \n
                                        • Improves vocabulary and semantic understanding.<\/li>\n
                                        • Strengthens verbal reasoning and logical thinking.<\/li>\n
                                        • Useful for competitive exams, puzzles, and language learning.<\/li>\n
                                        • Helps in categorizing knowledge and identifying relationships between concepts.<\/li>\n<\/ul>\n

                                          Examples<\/strong><\/p>\n

                                            \n
                                          • Group 1:<\/strong> {Rose, Lotus, Tulip, Sunflower} \u2192 All are flowers<\/li>\n
                                          • Group 2:<\/strong> {Dog, Cat, Cow, Horse} \u2192 All are domestic animals<\/li>\n<\/ul>\n

                                            Subtopics<\/strong><\/p>\n

                                            1. Synonym Groups<\/strong><\/p>\n

                                            Words that have similar meanings are grouped together.
                                            \nThis is useful for identifying semantic similarity.<\/p>\n

                                            Examples:<\/strong><\/p>\n

                                              \n
                                            • {Happy, Joyful, Cheerful, Glad} \u2192 All mean happiness<\/li>\n
                                            • {Big, Large, Huge, Enormous} \u2192 All mean large size<\/li>\n<\/ul>\n

                                              Odd-One-Out Example:<\/strong>
                                              \n{Fast, Quick, Rapid, Slow} \u2192 Slow (opposite meaning, not a synonym).<\/p>\n

                                              2. Category-Based Words<\/strong><\/p>\n

                                              Words are classified based on a specific category or theme.<\/p>\n

                                              Common Categories:<\/strong><\/p>\n

                                                \n
                                              • Fruits:<\/strong> Apple, Mango, Orange, Banana<\/li>\n
                                              • Vegetables:<\/strong> Carrot, Potato, Spinach, Onion<\/li>\n
                                              • Colors:<\/strong> Red, Blue, Green, Yellow<\/li>\n
                                              • Professions:<\/strong> Doctor, Engineer, Teacher, Lawyer<\/li>\n
                                              • Countries:<\/strong> India, Japan, Canada, Brazil<\/li>\n<\/ul>\n

                                                Example:<\/strong>
                                                \nGroup: {Red, Blue, Green, Yellow} \u2192 All are colors.
                                                \nOdd one out in {Lion, Tiger, Elephant, Mango} \u2192 Mango (not an animal).<\/p>\n

                                                3. Functional Groups<\/strong><\/p>\n

                                                Words are grouped based on function or usage.
                                                \nThis is very common in reasoning tests and requires thinking about what these words are used for.<\/p>\n

                                                Examples:<\/strong><\/p>\n

                                                  \n
                                                • Tools:<\/strong> Hammer, Screwdriver, Wrench, Pliers<\/li>\n
                                                • Vehicles:<\/strong> Car, Bus, Truck, Motorcycle<\/li>\n
                                                • Electronic Gadgets:<\/strong> Laptop, Smartphone, Tablet, Smartwatch<\/li>\n
                                                • Musical Instruments:<\/strong> Guitar, Piano, Violin, Flute<\/li>\n<\/ul>\n

                                                  Odd-One-Out Example:<\/strong>
                                                  \n{Car, Bike, Train, Spoon} \u2192 Spoon (not a vehicle).<\/p>\n

                                                  General Knowledge Classification<\/strong><\/p>\n

                                                  Definition<\/u><\/strong><\/p>\n

                                                  General Knowledge (GK) Classification is the process of grouping items based on real-world facts, awareness, and general information.
                                                  \nIt relies on factual knowledge rather than purely logical or mathematical rules.<\/p>\n

                                                  The goal is to identify the common theme or category among given items (e.g., sports, currencies, capitals) and use that to solve classification or odd-one-out questions.<\/p>\n

                                                  Importance of GK Classification<\/strong><\/p>\n

                                                    \n
                                                  • Improves awareness of the world.<\/li>\n
                                                  • Enhances quiz and competitive exam preparation.<\/li>\n
                                                  • Builds associative memory (connecting facts together).<\/li>\n
                                                  • Useful in reasoning tests, trivia games, and interviews.<\/li>\n<\/ul>\n

                                                    Examples<\/strong><\/p>\n

                                                      \n
                                                    • Group 1:<\/strong> {Dollar, Yen, Pound, Rupee} \u2192 All are currencies<\/li>\n
                                                    • Group 2:<\/strong> {Cricket, Football, Tennis, Hockey} \u2192 All are sports<\/li>\n<\/ul>\n

                                                      Subtopics<\/u><\/strong><\/p>\n

                                                      1. Countries & Capitals<\/strong><\/p>\n

                                                      Items are grouped based on geographical and political knowledge.<\/p>\n

                                                      Examples:<\/strong><\/p>\n

                                                        \n
                                                      • {Delhi, Paris, London, Tokyo} \u2192 All are capital cities<\/li>\n
                                                      • {India, Japan, Brazil, Canada} \u2192 All are countries<\/li>\n<\/ul>\n

                                                        Odd-One-Out Example:<\/strong>
                                                        \n{Paris, London, Tokyo, Amazon} \u2192 Amazon (not a capital city, it\u2019s a river\/region).<\/p>\n

                                                        2. Currencies & Symbols<\/strong><\/p>\n

                                                        Grouping based on national or international currencies and their symbols.<\/p>\n

                                                        Examples:<\/strong><\/p>\n

                                                          \n
                                                        • {Dollar (USD), Pound (GBP), Yen (JPY), Rupee (INR)} \u2192 All are currencies<\/li>\n
                                                        • {\u20ac, $, \u00a3, \u00a5} \u2192 All are currency symbols<\/li>\n<\/ul>\n

                                                          Odd-One-Out Example:<\/strong>
                                                          \n{Dollar, Yen, Bitcoin, Pound} \u2192 Bitcoin (cryptocurrency, not a national currency).<\/p>\n

                                                          3. Famous Personalities \/ Books<\/strong><\/p>\n

                                                          Grouping based on famous authors, leaders, scientists, or their works.<\/p>\n

                                                          Examples:<\/strong><\/p>\n

                                                            \n
                                                          • {Shakespeare, Dickens, Tolstoy, Hemingway} \u2192 All are authors<\/li>\n
                                                          • {Hamlet, Macbeth, Othello, Odyssey} \u2192 All are literary works (except Odyssey is by Homer, so could be odd-one-out depending on question).<\/li>\n<\/ul>\n

                                                            Odd-One-Out Example:<\/strong>
                                                            \n{Einstein, Newton, Edison, Picasso} \u2192 Picasso (artist, not scientist\/inventor).<\/p>\n

                                                            4. Inventions & Discoveries<\/strong><\/p>\n

                                                            Grouping based on scientific or technological contributions.<\/p>\n

                                                            Examples:<\/strong><\/p>\n

                                                              \n
                                                            • {Telephone, Radio, Television, Computer} \u2192 All are inventions<\/li>\n
                                                            • {Gravity, Penicillin, Electricity, Relativity} \u2192 All are discoveries\/theories<\/li>\n<\/ul>\n

                                                              Odd-One-Out Example:<\/strong>
                                                              \n{Telephone, Radio, Aeroplane, Beethoven} \u2192 Beethoven (composer, not inventor).<\/p>\n

                                                              Coding-Based Classification<\/strong><\/p>\n

                                                              Definition<\/u><\/strong><\/p>\n

                                                              Coding-Based Classification is the process of grouping coded words, letters, or numbers based on a common encryption, rule, or coding pattern.<\/p>\n

                                                              In these problems, words or numbers are represented in a coded form, and the task is to find the pattern used for the code and classify items accordingly.<\/p>\n

                                                              Importance of Coding-Based Classification<\/strong><\/p>\n

                                                                \n
                                                              • Strengthens logical reasoning and pattern recognition.<\/li>\n
                                                              • Frequently appears in competitive exams under \u201cCoding-Decoding\u201d questions.<\/li>\n
                                                              • Improves ability to detect hidden rules and relationships quickly.<\/li>\n
                                                              • Useful in puzzle-solving, cryptography basics, and mental ability tests.<\/li>\n<\/ul>\n

                                                                Example<\/strong><\/p>\n

                                                                If:<\/p>\n

                                                                  \n
                                                                • CAT = 3120<\/li>\n
                                                                • DOG = 4157<\/li>\n
                                                                • PIG = 1697<\/li>\n<\/ul>\n

                                                                  We can observe that each letter is converted into a number based on its position in the alphabet (A=1, B=2, …, Z=26) and then added or combined.
                                                                  \nThese words can be grouped based on having similar sum-of-positions patterns.<\/p>\n

                                                                  Subtopics<\/u><\/strong><\/p>\n

                                                                  1. Number Coding<\/strong><\/p>\n

                                                                  Numbers are assigned to letters or words according to a fixed rule.<\/p>\n

                                                                  Example:<\/strong>
                                                                  \nA=1, B=2, C=3…<\/p>\n

                                                                    \n
                                                                  • CAT \u2192 C(3) + A(1) + T(20) = 3120<\/li>\n
                                                                  • BAT \u2192 B(2) + A(1) + T(20) = 2120<\/li>\n<\/ul>\n

                                                                    Group Basis:<\/strong> All are coded using alphabet position values.<\/p>\n

                                                                    Odd-One-Out Example:<\/strong>
                                                                    \nIf BAT=2120, CAT=3120, RAT=18120, SUN=192114, the odd one is SUN (code is too long, may follow different pattern).<\/p>\n

                                                                    2. Letter Coding<\/strong><\/p>\n

                                                                    Letters are replaced with other letters according to a specific shift or pattern.<\/p>\n

                                                                    Example:<\/strong><\/p>\n

                                                                      \n
                                                                    • If A <\/strong>\u2192<\/strong> C, B <\/strong>\u2192<\/strong> D, C <\/strong>\u2192<\/strong> E<\/strong> (shift of +2),
                                                                      \n\tthen CAT becomes ECV.<\/li>\n<\/ul>\n

                                                                      Group Basis:<\/strong> Words following the same letter-shift rule.<\/p>\n

                                                                      Odd-One-Out Example:<\/strong>
                                                                      \nIf CAT=ECV, DOG=FQI, PIG=RKI, and BAT=CFV, the odd one is BAT (does not follow +2 shift).<\/p>\n

                                                                      3. Substitution Coding<\/strong><\/p>\n

                                                                      Words are replaced with other words or symbols based on a predefined dictionary-like code.<\/p>\n

                                                                      Example:<\/strong><\/p>\n

                                                                        \n
                                                                      • \u2018APPLE\u2019 is coded as \u2018BANANA\u2019<\/li>\n
                                                                      • \u2018MANGO\u2019 is coded as \u2018ORANGE\u2019<\/li>\n<\/ul>\n

                                                                        Here, the classification is based on pairing substitution.<\/p>\n

                                                                        Group Basis:<\/strong> All codes represent fruit names through substitution.<\/p>\n

                                                                        4. Mixed Coding<\/strong><\/p>\n

                                                                        A combination of numbers, letters, and substitutions is used simultaneously.<\/p>\n

                                                                        Example:<\/strong><\/p>\n

                                                                          \n
                                                                        • RED = 18E4 (R=18, E kept same, D=4)<\/li>\n
                                                                        • BLUE = 2L21E (B=2, L kept same, U=21, E kept same)<\/li>\n<\/ul>\n

                                                                          Group Basis:<\/strong> Codes that mix numbers + letters but follow a consistent logic.<\/p>\n

                                                                          Example: –<\/u><\/strong><\/p>\n

                                                                          Examine the seven items below. Exactly one item does NOT belong<\/strong> with the rest. Identify it and give a rigorous justification (show three independent reasons<\/em> \u2014 one from each of these domains: (A) letter\/alphabet patterning (positions or sequences), (B) numeric\/coding property, (C) semantic \/ general-knowledge category<\/strong>).<\/p>\n

                                                                          Items:<\/p>\n

                                                                            \n
                                                                          1. MANGO \u2014 13\u00b71\u00b714\u00b77\u00b715 <\/strong>\u2192<\/strong> 13114715<\/strong><\/li>\n
                                                                          2. BANANA \u2014 2\u00b71\u00b714\u00b71\u00b714\u00b71 <\/strong>\u2192<\/strong> 21114141<\/strong><\/li>\n
                                                                          3. ORANGE \u2014 15\u00b718\u00b71\u00b714\u00b77\u00b75 <\/strong>\u2192<\/strong> 151811475<\/strong><\/li>\n
                                                                          4. PAPAYA \u2014 16\u00b71\u00b716\u00b71\u00b725\u00b71 <\/strong>\u2192<\/strong> 161161251<\/strong><\/li>\n
                                                                          5. TOMATO \u2014 20\u00b715\u00b713\u00b71\u00b720\u00b715 <\/strong>\u2192<\/strong> 20151312015<\/strong><\/li>\n
                                                                          6. CARROT \u2014 3\u00b71\u00b718\u00b718\u00b715\u00b720 <\/strong>\u2192<\/strong> 318181520<\/strong><\/li>\n
                                                                          7. BANOFFEE \u2014 2\u00b71\u00b714\u00b715\u00b76\u00b76\u00b75\u00b75 <\/strong>\u2192<\/strong> 2114156655<\/strong><\/li>\n<\/ol>\n

                                                                            (Each arrow shows the raw letter-position concatenation, i.e. A=1, B=2, …, Z=26. The numbers are presented as concatenations<\/em> of positions, not sums.)<\/p>\n

                                                                            Question:<\/strong> Which one is the odd item out? Prove it by giving three independent reasons (alphabet\/position pattern, numeric\/coding property, and semantic\/GK property) that together rule it out.<\/p>\n

                                                                            Solutio<\/u>n: –<\/strong><\/p>\n

                                                                            (A) Letter \/ Alphabet pattern (position & parity test)<\/strong><\/p>\n

                                                                            Observation rule used for comparison:<\/strong> for each item, consider the parity pattern (even\/odd) of the sequence of letter positions<\/em> and whether that parity sequence is a palindrome (reads same forward\/back).<\/p>\n

                                                                            Compute parity sequences (E=even, O=odd):<\/p>\n

                                                                              \n
                                                                            1. MANGO<\/strong> positions: 13(O), 1(O), 14(E), 7(O), 15(O) \u2192 parity: O O E O O \u2192 palindrome<\/strong> (reads same reversed).<\/li>\n
                                                                            2. BANANA<\/strong>: 2(E),1(O),14(E),1(O),14(E),1(O) \u2192 E O E O E O \u2192 palindrome<\/strong>.<\/li>\n
                                                                            3. ORANGE<\/strong>: 15(O),18(E),1(O),14(E),7(O),5(O) \u2192 O E O E O O \u2192 reversed O O E O E O \u2192 not palindrome<\/strong>? Wait \u2014 check carefully: forward O E O E O O, reverse O O E O E O \u2014 they differ \u2192 not palindrome<\/strong>. (Keep this in memory.)<\/li>\n
                                                                            4. PAPAYA<\/strong>: 16(E),1(O),16(E),1(O),25(O),1(O) \u2192 E O E O O O \u2192 reversed O O O E O E \u2192 different \u2192 not palindrome<\/strong>.<\/li>\n
                                                                            5. TOMATO<\/strong>: 20(E),15(O),13(O),1(O),20(E),15(O) \u2192 E O O O E O \u2192 reversed O E O O O E \u2192 not palindrome<\/strong>.<\/li>\n
                                                                            6. CARROT<\/strong>: 3(O),1(O),18(E),18(E),15(O),20(E) \u2192 O O E E O E \u2192 reversed E O E E O O \u2192 not palindrome<\/strong>.<\/li>\n
                                                                            7. BANOFFEE<\/strong>: 2(E),1(O),14(E),15(O),6(E),6(E),5(O),5(O) \u2192 E O E O E E O O \u2192 reversed O O E E O E O E \u2192 not the same<\/strong>.<\/li>\n<\/ol>\n

                                                                              So at first glance palindromicity alone does not single one out uniquely (only MANGO & BANANA are palindromes). But we can make a stronger alphabetic pattern test<\/em>:<\/p>\n

                                                                              Test A-2: Count of consecutive same-parity runs<\/strong>
                                                                              \nCount how many times the parity changes<\/em> as you move left\u2192right (number of transitions). For each:<\/p>\n

                                                                                \n
                                                                              • MANGO (O O E O O): transitions at 2\u21923 (O\u2192E), 3\u21924 (E\u2192O) = 2 transitions<\/strong>.<\/li>\n
                                                                              • BANANA (E O E O E O): transitions each adjacent pair \u2192 five transitions = 5<\/strong>.<\/li>\n
                                                                              • ORANGE (O E O E O O): transitions at many = 4<\/strong>.<\/li>\n
                                                                              • PAPAYA: transitions = 3<\/strong>.<\/li>\n
                                                                              • TOMATO: transitions = 4<\/strong>.<\/li>\n
                                                                              • CARROT: transitions = 4<\/strong>.<\/li>\n
                                                                              • BANOFFEE (E O E O E E O O): transitions at 1\u21922,2\u21923,3\u21924,4\u21925,5\u21926(no change),6\u21927,7\u21928 = 6<\/strong> transitions.<\/li>\n<\/ul>\n

                                                                                Conclusion (A):<\/strong> BANOFFEE has the maximum number of parity transitions (6)<\/strong> among the seven, and no other item reaches 6<\/em>. BANOFFEE therefore uniquely fails the \u201clow transition \/ grouped parity\u201d pattern that all the others approximate (others have 2\u20135 transitions). This shows BANOFFEE is atypical in a strict alphabet \/ parity-pattern metric.<\/p>\n

                                                                                (So BANOFFEE fails an alphabet\/position structural test that the others pass or at least do not maximize.)<\/p>\n

                                                                                (B) Numeric \/ Coding property (concatenated positions and factorization test)<\/strong><\/p>\n

                                                                                We used concatenation of letter positions<\/em> as the given coding. Now use this numeric test:<\/p>\n

                                                                                Test B-1 (length of concatenated code in digits):<\/strong>
                                                                                \nCompute number of digits in the concatenated code (note two-digit letters e.g. 16, 20 produce two digits). Count digits:<\/p>\n

                                                                                  \n
                                                                                1. MANGO: 13(2) +1(1)+14(2)+7(1)+15(2) = 8 digits<\/strong> \u2192 13114715 (8 digits).<\/li>\n
                                                                                2. BANANA: 2+1+14+1+14+1 = 1+1+2+1+2+1 = 8 digits<\/strong> \u2192 21114141.<\/li>\n
                                                                                3. ORANGE: 15(2)+18(2)+1(1)+14(2)+7(1)+5(1) = 2+2+1+2+1+1 = 9 digits<\/strong> \u2192 151811475.<\/li>\n
                                                                                4. PAPAYA: 16(2)+1+16(2)+1+25(2)+1 = 2+1+2+1+2+1 = 9 digits<\/strong> \u2192 161161251.<\/li>\n
                                                                                5. TOMATO: 20(2)+15(2)+13(2)+1+20(2)+15(2) = 2+2+2+1+2+2 = 11 digits<\/strong> \u2192 20151312015.<\/li>\n
                                                                                6. CARROT: 3+1+18+18+15+20 = 1+1+2+2+2+2 = 10 digits<\/strong> \u2192 318181520.<\/li>\n
                                                                                7. BANOFFEE: 2+1+14+15+6+6+5+5 = 1+1+2+2+1+1+1+1 = 10 digits<\/strong> \u2192 2114156655.<\/li>\n<\/ol>\n

                                                                                  So digit-lengths are: {8,8,9,9,11,10,10}. No unique immediate singleton by length<\/strong> (two items share 10 digits, etc.)<\/p>\n

                                                                                  Test B-2 (divisibility pattern \/ prime-digit sum):
                                                                                  \nCompute the digit-sum (sum of digits of the concatenated number) and then test if that digit-sum is a prime<\/em>.<\/p>\n

                                                                                  Quick, careful digit sums (compute digit by digit):<\/p>\n

                                                                                    \n
                                                                                  1. MANGO 13114715<\/strong>: digits 1+3+1+1+4+7+1+5 = 23 \u2192 23 is prime<\/strong>.<\/li>\n
                                                                                  2. BANANA 21114141<\/strong>: 2+1+1+1+4+1+4+1 = 15 \u2192 15 not prime<\/strong>.<\/li>\n
                                                                                  3. ORANGE 151811475<\/strong>: 1+5+1+8+1+1+4+7+5 = 33 \u2192 33 not prime<\/strong>.<\/li>\n
                                                                                  4. PAPAYA 161161251<\/strong>: 1+6+1+1+6+1+2+5+1 = 24 \u2192 24 not prime<\/strong>.<\/li>\n
                                                                                  5. TOMATO 20151312015<\/strong>: digits sum = 2+0+1+5+1+3+1+2+0+1+5 = 21 \u2192 21 not prime<\/strong>.<\/li>\n
                                                                                  6. CARROT 318181520<\/strong>: 3+1+8+1+8+1+5+2+0 = 29 \u2192 29 is prime<\/strong>.<\/li>\n
                                                                                  7. BANOFFEE 2114156655<\/strong>: 2+1+1+4+1+5+6+6+5+5 = 36 \u2192 36 not prime<\/strong>.<\/li>\n<\/ol>\n

                                                                                    Conclusion (B):<\/strong> Exactly two items (MANGO and CARROT) have prime digit-sum (23 and 29). Several are composite. But note: BANOFFEE’s digit-sum = 36 (a perfect square) while among the seven it is the only one whose digit-sum is a perfect square (36 = 6\u00b2). Check other digit sums: 15,33,24,21,29,36,23 \u2014 only 36 is a perfect square. That\u2019s a clean numeric distinguisher:<\/p>\n

                                                                                      \n
                                                                                    • BANOFFEE \u2192 digit-sum 36 = 6\u00b2<\/strong> (unique perfect square).
                                                                                      \n\tTherefore BANOFFEE is unique by this numeric\/coding property as well.<\/li>\n<\/ul>\n

                                                                                      (C) Semantic \/ General Knowledge classification<\/strong><\/p>\n

                                                                                      Test C-1 (semantic category):<\/strong> Are these words fruits, vegetables, or desserts\/derived foods?<\/p>\n

                                                                                        \n
                                                                                      • MANGO \u2014 fruit (tropical fruit).<\/li>\n
                                                                                      • BANANA \u2014 fruit.<\/li>\n
                                                                                      • ORANGE \u2014 fruit.<\/li>\n
                                                                                      • PAPAYA \u2014 fruit.<\/li>\n
                                                                                      • TOMATO \u2014 botanically a fruit, culinarily used as a vegetable (ambiguous).<\/li>\n
                                                                                      • CARROT \u2014 vegetable (root vegetable).<\/li>\n
                                                                                      • BANOFFEE \u2014 not a raw produce item: Banoffee<\/strong> is an English dessert<\/strong> (banoffee pie \u2014 banana + toffee). It is not<\/em> the name of a raw fruit nor a vegetable.<\/li>\n<\/ul>\n

                                                                                        So semantically, six items are names of primary agricultural produce<\/em> (fruits or vegetables or botanically fruit), while BANOFFEE is a processed food \/ dessert (a compound food word), not a single raw produce item. That already separates it.<\/p>\n

                                                                                        Test C-2 (botanical vs culinary):<\/strong> All other six items refer to single botanical species\/names (Mango\u2014Mangifera indica, Banana\u2014Musa spp., Orange\u2014Citrus \u00d7 sinensis, Papaya\u2014Carica papaya, Tomato\u2014Solanum lycopersicum, Carrot\u2014Daucus carota). BANOFFEE<\/strong> is not a species \u2014 it’s a composite dessert name<\/strong> (banana + toffee). So it fails the species \/ botanical-name membership property.<\/p>\n

                                                                                        Thus on semantic\/GK grounds BANOFFEE is clearly different.<\/p>\n

                                                                                        Because these three distinguishing tests are independent<\/strong> (alphabet pattern, numeric property of the code, and real-world semantic category), BANOFFEE is definitively the odd one out.<\/p>\n

                                                                                        \u00a0<\/p>\n

                                                                                        \u00a0<\/p>\n

                                                                                        <\/p>\n