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CAT Quantitative Aptitude | CAT Permutations and Combinations, Probability

CAT Questions / Permutation & Probability

CAT Permutation and Combination question that appears in the Quantitative Aptitude section of the CAT Exam broadly tests an aspirant on the concepts - Permutation, Combination, Probability, Counting and so on. In CAT Exam, one can generally expect to get 2~3 questions from CAT Permutation and Combination and Probability. CAT Permutation and Combination and Probability is an important topic in the CAT Exam. Make use of 2IIMs Free CAT Questions, provided with detailed solutions and Video explanations to obtain a wonderful CAT score. If you would like to take these questions as a Quiz, head on here to take these questions in a test format, absolutely free.

  1. CAT Permutation and Combination: Counting natural numbers

    Sum of three Natural numbers a, b and c is 10. How many ordered triplets (a, b, c) exist?

    1. 45
    2. 36
    3. 54
    4. 28

    Choice B
    36

  2. CAT Permutation and Combination: Counting Whole Numbers

    Sum of three Whole numbers a, b and c is 10. How many ordered triplets (a, b, c) exist?

    1. 66
    2. 78
    3. 72
    4. 56

    Choice A
    66

  3. CAT Permutation and Combination: Counting - Toys and Boxes

    In how many ways 11 identical toys be placed in 3 distinct boxes such that no box is empty?

    1. 72
    2. 54
    3. 45
    4. 36

    Choice C
    45

  4. CAT Permutation and Combination: Puzzle

    a, b, c are three distinct integers from 2 to 10 (both inclusive). Exactly one of ab, bc and ca is odd. abc is a multiple of 4. The arithmetic mean of a and b is an integer and so is the arithmetic mean of a, b and c. How many such triplets are possible (unordered triplets)?

    1. 8
    2. 6
    3. 2
    4. 4

    Choice D
    4

  5. CAT Permutation and Combination: Counting 7 Digit Numbers

    A seven-digit number comprises of only 2's and 3's. How many of these are multiples of 12?

    1. 11
    2. 12
    3. 10
    4. 22

    Choice A
    11

  6. CAT Permutation and Combination: Alphabetical Order

    If all words with 2 distinct vowels and 3 distinct consonants were listed alphabetically, what would be the rank of “ACDEF’?

    1. 4716
    2. 4720
    3. 4718
    4. 1717

    Choice C
    4718

  7. CAT Permutation and Combination: Basics

    If we listed all numbers from 100 to 10,000, how many times would the digit 3 be printed?

    1. 3980
    2. 3700
    3. 3840
    4. 3780

    Choice A
    3980

  8. CAT Permutation and Combination: Number System

    From the digits 2, 3, 4, 5, 6 and 7, how many 5-digit numbers can be formed that have distinct digits and are multiples of 12?

    1. 36
    2. 60
    3. 84
    4. 72

    Choice B
    60

  9. CAT Permutation and Combination: Numbers in Different Bases

    All numbers from 1 to 200 (in decimal system) are written in base 6 and base 7 systems. How many of the numbers will have a non-zero units digit in both base 6 and base 7 notations?

    1. 143
    2. 200
    3. 157
    4. 122

    Choice A
    143

  10. CAT Permutation and Combination: Non-Zero Numbers

    All numbers from 1 to 150 (in decimal system) are written in base 6 notation. How many of these will contain zero's?

    1. 25
    2. 20
    3. 35
    4. 45

    Choice D
    45

  11. CAT Permutation and Combination: 5 Digit Numbers

    How many numbers of up to 5 digits can be created using the digits 1, 2, 3 and 5 each at least once such that they are a multiple of 15?

    1. 24
    2. 18
    3. 15
    4. 12

    Choice D
    12

  12. CAT Permutation and Combination: Numbers with Distinct Digits

    How many odd numbers with distinct digits can be created using the digits 1, 2, 3, 4, 5 and 6?

    1. 975
    2. 960
    3. 978
    4. 986

    Choice C
    978

  13. CAT Permutation and Combination: Alphabetical Order

    All the rearrangements of the word "DEMAND" are written without including any word that has two D's appearing together. If these are arranged alphabetically, what would be the rank of "DEMAND"?

    1. 36
    2. 74
    3. 42
    4. 86

    Choice B
    74

  14. CAT Probability: GP

    A and B take part in a duel. A can strike with an accuracy of 0.6. B can strike with an accuracy of 0.8. A has the first shot, post which they strike alternately. What is the probability that A wins the duel?

    1. \\frac{7}{10}\\)
    2. \\frac{15}{23}\\)
    3. \\frac{2}{3}\\)
    4. \\frac{11}{17}\\)

    Choice B
    \\frac{15}{23}\\)

  15. CAT Probability: Bayes Theorem

    Doctors have devised a test for leptospirosis that has the following property: For any person suffering from lepto, there is a 90% chance of the test returning positive. For a person not suffering from lepto, there is an 80% chance of the test returning negative. It is known that 10% of people who go for testing have lepto. If a person who gets tested gets a +ve result for lepto (as in, the test result says they have got lepto), what is the probability that they actually have lepto?

    1. \\frac{7}{10}\\)
    2. \\frac{8}{11}\\)
    3. \\frac{1}{3}\\)
    4. \\frac{1}{2}\\)

    Choice C
    \\frac{1}{3}\\)

  16. CAT Permutation and Combination: Rearranging Letters

    In how many ways can letters the word ATTITUDE be rearranged such that no two Ts are adjacent to each other?

    1. 6720
    2. 2400
    3. 4320
    4. 1800

    Choice B
    2400

  17. CAT Permutation and Combination: Possible Integer Solutions

    2a + 5b = 103. How many pairs of positive integer values can a, b take such that a > b?

    1. 7
    2. 9
    3. 14
    4. 15

    Choice A
    7

  18. CAT Probability: Rearrangement of Letters

    If all the rearrangements of the word AMAZON are considered, what is the probability that M will feature between the 2As?

    1. \\frac{1}{3}\\)
    2. \\frac{1}{6}\\)
    3. \\frac{2}{5}\\)
    4. \\frac{3}{8}\\)

    Choice A
    \\frac{1}{3}\\)

  19. CAT Probability: Number Theory

    N is a 3-digit number that is a multiple of 7; what is the probability that it will be a multiple of 5?

    1. \\frac{1}{5}\\)
    2. \\frac{11}{54}\\)
    3. \\frac{13}{64}\\)
    4. \\frac{13}{66}\\)

    Choice C
    \\frac{13}{64}\\)

  20. CAT Probability

    A boss decides to distribute Rs. 2000 between 2 employees. He knows X deserves more that Y, but does not know how much more. So he decides to arbitrarily break Rs. 2000 into two parts and give X the bigger part. What is the chance that X gets twice as much as Y or more?

    1. \\frac{2}{5}\\)
    2. \\frac{1}{2}\\)
    3. \\frac{1}{3}\\)
    4. \\frac{2}{3}\\)

    Choice D
    \\frac{2}{3}\\)

  21. CAT Permutation and Combination: Rolling a Die

    I roll a die four times. In how many outcomes do we have two throws have the same number and the other two something different?

  22. CAT Permutation and Combination: Card Pack - Selection

    In how many ways can be select 5 cards from a card pack such that all 4 suits appear?

    1. 52728
    2. 405646
    3. 685464
    4. 4056

    Choice C
    685464

  23. CAT Permutation and Combination: Rolling a Die

    I roll a die 4 times. In how many outcomes will each subsequent throw be greater that the previous one?

    1. 15
    2. 48
    3. 30
    4. 60

    Choice A
    15

  24. CAT Permutation and Combination: 3 Digit Number

    Find 3 digit numbers such that product of their digits is a natural number less than 5?

    1. 11
    2. 15
    3. 13
    4. 17

    Choice C
    13

  25. CAT Permutation and Combination: 3 Digit Number - Sum of Digits

    Find all 3 digit numbers such that sum of their digits is a whole number less than 5?

    1. 18
    2. 20
    3. 19
    4. 17

    Choice B
    20

  26. CAT Permutation and Combination: Sum of Rearrangements

    What is sum of all rearrangements of the 4-digit number 3214?

    1. 66660
    2. 55554
    3. 60048
    4. 65024

    Choice A
    66660

  27. CAT Permutation and Combination: Sum of Rearrangements

    What is sum of all rearrangements of the 4-digit number 3321?

    1. 29999
    2. 27777
    3. 28888
    4. 29997

    Choice D
    29997

  28. CAT Permutation and Combination: Forming Triangles

    Of 22 points on a plane, 8 are on a straight line, 7 are on another straight line and 10 are on a third straight line. How many triangles can be drawn by connecting some three points from these 22?

    1. 22C3
    2. 22C3 - (8C3+ 7C3 +10C3)
    3. 22C3 + (8C3+ 7C3 +10C3)
    4. 8C3+ 7C3 +10C3

    Choice B
    22C3 - (8C3+ 7C3 +10C3

  29. CAT Probability: Selection

    A bag contains 4 red and 3 black balls. A second bag contains 2 red and 3 black balls. One bag is selected at random. If from the selected bag one ball is drawn, then what is the probability that the ball drawn is red?

    1. \\frac{39}{70}\\)
    2. \\frac{41}{70}\\)
    3. \\frac{29}{70}\\)
    4. \\frac{17}{35}\\)

    Choice D
    \\frac{17}{35}\\)

  30. CAT Permutation and Combination: MANANA Rearrangment

    In how many ways can we rearrange the letters of the word MANANA such that no two A’s are adjacent to each other?

    1. 3 ! * 5C3
    2. 4 ! * 4C3
    3. 3 * 4C3
    4. 3 ! * 4C3

    Choice C
    3 * 4C3

  31. CAT Permutation and Combination: I between 2 A's

    How many ways are there for arranging letters of the word AMAZING such that the ‘I’ appears between the two ‘A’s?

    1. 5 ! ways
    2. 7 ! ways
    3. 8 ! ways
    4. 4 ! ways

    Choice A
    5 ! ways

  32. CAT Permutation and Combination: Room Allocation

    In how many ways can 6 boys be allotted into 5 rooms such that no room is empty and all 6 boys are accommodated?

    1. 6 * 5 ! ways
    2. 7 * 5 ! ways
    3. 3 * 3 ! ways
    4. 15 * 5 ! ways

    Choice D
    15 * 5 ! ways

  33. CAT Permutation and Combination: 3 Digit numbers > 500

    How many 3-digit numbers greater than 500 contain the digit 9 appearing at least once?

    1. 191
    2. 176
    3. 153
    4. 189

    Choice B
    176

The Questions that follow, are from actual CAT papers. If you wish to take them separately or plan to solve actual CAT papers at a later point in time, It would be a good idea to stop here.

  1. CAT 2023 Slot 1 - QA

    The number of all natural numbers up to 1000 with non-repeating digits is

    1. 585
    2. 504
    3. 648
    4. 738

    Choice D
    738

  2. CAT 2022 Slot 2 - QA

    The number of integers greater than 2000 that can be formed with the digits 0, 1, 2, 3, 4, 5, using each digit at most once, is

    1. 1440
    2. 1200
    3. 1420
    4. 1480

    Choice A
    1440

  3. CAT 2022 Slot 2 - QA

    In an examination, there were 75 questions. 3 marks were awarded for each correct answer, 1 mark was deducted for each wrong answer and 1 mark was awarded for each unattempted question. Rayan scored a total of 97 marks in the examination. If the number of unattempted questions was higher than the number of attempted questions, then the maximum number of correct answers that Rayan could have given in the examination is

      24

    1. CAT 2022 Slot 1 - QA

      The number of ways of distributing 20 identical balloons among 4 children such that each child gets some balloons but no child gets an odd number of balloons, is

        84

      1. CAT 2021 Slot 2 - QA

        The number of ways of distributing 15 identical balloons, 6 identical pencils and 3 identical erasers among 3 children, such that each child gets at least four balloons and one pencil, is

          1000

          Correct: 8.45%
          Incorrect: 41.39%
          Unattempted: 50.16%

        1. CAT 2021 Slot 1 - QA

          The number of groups of three or more distinct numbers that can be chosen from 1, 2, 3, 4, 5, 6, 7 and 8 so that the groups always include 3 and 5, while 7 and 8 are never included together is

            47

            Correct: 1.83%
            Incorrect: 40.1%
            Unattempted: 58.07%

          1. CAT 2020 Question Paper Slot 3 - Combinatorics

            How many integers in the set {100, 101, 102, ..., 999} have at least one digit repeated?

            Correct Answer: 252

          2. CAT 2018 Question Paper Slot 2 - Permutation & Combination

            In a tournament, there are 43 junior level and 51 senior level participants. Each pair of juniors play one match. Each pair of seniors play one match. There is no junior versus senior match. The number of girl versus girl matches in junior level is 153, while the number of boy versus boy matches in senior level is 276. The number of matches a boy plays against a girl is (TITA)

            1098

          3. CAT 2018 Question Paper Slot 1 - Number theory

            How many numbers with two or more digits can be formed with the digits 1, 2, 3, 4, 5, 6, 7, 8, and 9 so that in every such number, each digit is used at most once and the digits appear in the ascending order?[TITA]

            502

          4. CAT 2017 Question Paper Slot 2 - Permutation & Combination

            In how many ways can 8 identical pens be distributed among Amal, Bimal, and Kamal so that Amal gets at least 1 pen, Bimal gets at least 2 pens, and Kamal gets at least 3 pens? [TITA]

            6

          5. CAT 2017 Question Paper Slot 2 - Permutation & Combination

            How many four digit numbers, which are divisible by 6, can be formed using the digits 0, 2, 3, 4, 6, such that no digit is used more than once and 0 does not occur in the left-most position? [TITA]

            50

          6. CAT 2017 Question Paper Slot 1 - Permutation & Combination

            The number of solutions (x, y, z) to the equation x – y – z = 25, where x, y, and z are positive integers such that x ≤ 40, y ≤ 12, and z ≤ 12 is

            1. 101
            2. 99
            3. 87
            4. 105

            Choice B
            99

          7. CAT 2017 Question Paper Slot 1 - Permutation & Combination

            Let AB, CD, EF, GH, and JK be five diameters of a circle with center at O. In how many ways can three points be chosen out of A, B, C, D, E, F, G, H, J, K, and O so as to form a triangle? (TITA)

            160

          8. CAT 2017 Question Paper Slot 1 - Permutation & Combination

            In how many ways can 7 identical erasers be distributed among 4 kids in such a way that each kid gets at least one eraser but nobody gets more than 3 erasers?

            1. 16
            2. 20
            3. 14
            4. 15

            Choice A
            16

          The Questions that follow, are from actual XAT papers. If you wish to take them separately or plan to solve actual XAT papers at a later point in time, It would be a good idea to stop here.

          1. XAT 2020 Question Paper - QADI

            Ashok has a bag containing 40 cards, numbered with the integers from 1 to 40. No two cards are numbered with the same integer. Likewise, his sister Shilpa has another bag containing only five cards that are numbered with the integers from 1 to 5, with no integer repeating. Their mother, Latha, randomly draws one card each from Ashok’s and Shilpa’s bags and notes down their respective numbers. If Latha divides the number obtained from Ashok’s bag by the number obtained from Shilpa’s, what is the probability that the remainder will not be greater than 2?

            1. 0.91
            2. 0.87
            3. 0.94
            4. 0.73
            5. 0.8

            Choice B
            0.87

          2. XAT 2020 Question Paper - QADI

            A box contains 6 cricket balls, 5 tennis balls and 4 rubber balls. Of these, some balls are defective. The proportion of defective cricket balls is more than the proportion of defective tennis balls but less than the proportion of defective rubber balls. Moreover, the overall proportion of defective balls is twice the proportion of defective tennis balls. What BEST can be said about the number of defective rubber balls in the box?

            1. It is exactly 3
            2. It is either 3 or 4
            3. It is exactly 2
            4. It is either 2 or 3
            5. It is either 0 or 1

            Choice A
            It is exactly 3

          3. XAT 2019 Question Paper - QADI

            A bag contains marbles of three colours-red, blue and green. There are 8 blue marbles in the bag.There are two additional statement of facts available:
            1. If we pull out marbles from the bag at random, to guarantee that we have at least 3 green marbles, we need to extract 17 marbles.
            2. If we pull out marbles from the bag at random, to guarantee that we have at least 2 red marbles, we need to extract 19 marbles.

            Which of the two statements above, alone or in combination shall be sufficient to answer the question "how many green marbles are there in the bag"?

            1. Statement 1 alone is sufficient, but statement 2 alone is not sufficient to answer the question.
            2. Statement 2 alone is sufficient, but statement 1 alone is not sufficient to answer the question.
            3. Statements 1 and 2 together are not sufficient, and additional data is needed to answer the question.
            4. Each statement alone is sufficient to answer the question.
            5. Both statements taken together are sufficient to answer the question, but neither statement alone is sufficient.

            Choice B
            Statement 2 alone is sufficient, but statement 1 alone is not sufficient to answer the question.

          4. XAT 2018 Question Paper - QADI

            A coin of radius 3 cm is randomly dropped on a square floor full of square shaped tiles of side 10 cm each. What is the probability that the coin will land completely within a tile? In other words, the coin should not cross the edge of any tile.

            1. 0.91
            2. 0.5
            3. 0.49
            4. 0.36
            5. 0.16

            Choice E
            0.16

          The Questions that follow, are from actual IPMAT papers. If you wish to take them separately or plan to solve actual IPMAT papers at a later point in time, It would be a good idea to stop here.

          1. IPMAT 2020 Sample Paper - IPM Rohtak Quants

            A bag contains 4 blue, 5 white and 6 green balls. Two balls are drawn at random. What is the probability that one ball is white?

            1. \\frac{10}{21}\\)
            2. \\frac{1}{2}\\)
            3. \\frac{3}{4}\\)
            4. \\frac{2}{35}\\)

            Choice D
            \\frac{2}{35}\\)

          2. IPMAT 2020 Question Paper - IPM Rohtak Quants

            Five racquets needs to be placed in three boxes. Each box can hold all the five racquets. In how many ways can the racquets be placed in the boxes so that no box can be empty if all racquets are different but all boxes are identical?

            1. 24
            2. 25
            3. 27
            4. 26

            Choice B
            25

          3. IPMAT 2020 Question Paper - IPM Rohtak Quants

            How many signposts can be made using 6 different coloured symbols when any number of them can be posted at a time?

            1. 1988
            2. 1976
            3. 1966
            4. 1956

            Choice D
            1956

          4. IPMAT 2020 Question Paper - IPM Rohtak Quants

            There are 12 copies of Beetles CDs, 7 copies of Pink Floyd CDs, 3 different CDs of Michael Jackson, and 2 different CDs of Madonna. Find the number of ways in which one or more than one CD can be selected?

            1. 3388
            2. 3376
            3. 3366
            4. 3327

            Choice D
            3327

          5. IPMAT 2020 Question Paper - IPM Rohtak Quants

            Please calculate in how many ways can a platoon of sixteen soldiers be chosen out of a total of twenty soldiers for a surgical strike?

            1. 4845
            2. 4800
            3. 4855
            4. 4955

            Choice A
            4845

          6. IPMAT 2020 Question Paper - IPM Rohtak Quants

            A box contains 2 white shirts, 3 black shirts, and 4 red shirts. In how many ways can 3 shirts be drawn from the box, if at least one black shirt is to be included in the draw?

            1. 60
            2. 40
            3. 64
            4. 82

            Choice C
            64

          7. IPMAT 2020 Question Paper - IPM Rohtak Quants

            Can you calculate in how many ways can 7 tennis players can be seated in a circular order?

            1. 330
            2. 730
            3. 720
            4. 820

            Choice C
            720

          8. IPMAT 2020 Question Paper - IPM Rohtak Quants

            Please calculate in how many ways can a set of five players be formed out of a total of ten players such that two particular players should be involved in each set?

            1. 60
            2. 72
            3. 56
            4. 75

            Choice C
            56

          9. IPMAT 2020 Question Paper - IPM Indore Quants

            out of 13 objects, 4 are indistinguishable and rest are distinct. The number of ways we can choose 4 objects out of 13 objects is __________.

            256

          10. IPMAT 2020 Question Paper - IPM Indore Quants

            The probability that a randomly chosen factor of 1019 is a multiple of 1015 is

            1. \\frac{1}{25})
            2. \\frac{1}{12})
            3. \\frac{1}{20})
            4. \\frac{1}{16})

            Choice C
            \\frac{1}{16})

          11. IPMAT 2020 Question Paper - IPM Indore Quants

            A man is known to speak the truth on an average 4 out of 5 times. He throws a die and reports that it is a five. The probability that it is actually a five is

            1. \\frac{4}{9})
            2. \\frac{5}{9})
            3. \\frac{4}{15})
            4. \\frac{2}{15})

            None of these!
            \\frac{4}{5})

          12. IPMAT 2019 Question Paper - IPM Indore Quants

            From a pack of 52 cards, we draw one by one, without replacement. If f(n) is the probability that an Ace will appear at the nth turn, then

            1. f(2) = \\frac{1}{13}\\) > f(3)
            2. \\frac{1}{13}\\) > f(2) > f(3)
            3. f(3) > f(2) = \\frac{1}{13}\\)
            4. f(2) = f(3) = \\frac{1}{13}\\)

            Choice D
            f(2) = f(3) = \\frac{1}{13}\\)

          13. IPMAT 2019 Question Paper - IPM Indore Quants

            A die is thrown three times and the sum of the three numbers is found to be 15. The probability that the first throw was a four is

            1. \\frac{1}{6}\\)
            2. \\frac{1}{4}\\)
            3. \\frac{1}{5}\\)
            4. \\frac{1}{10}\\)

            Choice C
            \\frac{1}{5}\\)

          14. IPMAT 2019 Question Paper - IPM Indore Quants

            In a given village there are only three sizes of families: families with 2 members, families with 4 members and families with 6 members. The proportion of families with 2,4 and 6 members are roughly equal. A poll is conducted in this village wherein a person is chosen at random and asked about his/her family size. The average family size computed by sampling 1000 such persons from the village would be closest to

            1. 4
            2. 4.667
            3. 4.333
            4. 3.667

            Choice B
            4.667

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          Phone: +128413562823324

          Job: IT Strategist

          Hobby: Video gaming, Basketball, Web surfing, Book restoration, Jogging, Shooting, Fishing

          Introduction: My name is Rev. Porsche Oberbrunner, I am a zany, graceful, talented, witty, determined, shiny, enchanting person who loves writing and wants to share my knowledge and understanding with you.