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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.

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?
 45
 36
 54
 28
Choice B
36 
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?
 66
 78
 72
 56
Choice A
66 
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?
 72
 54
 45
 36
Choice C
45 
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)?
 8
 6
 2
 4
Choice D
4 
CAT Permutation and Combination: Counting 7 Digit Numbers
A sevendigit number comprises of only 2's and 3's. How many of these are multiples of 12?
 11
 12
 10
 22
Choice A
11 
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’?
 4716
 4720
 4718
 1717
Choice C
4718 
CAT Permutation and Combination: Basics
If we listed all numbers from 100 to 10,000, how many times would the digit 3 be printed?
 3980
 3700
 3840
 3780
Choice A
3980 
CAT Permutation and Combination: Number System
From the digits 2, 3, 4, 5, 6 and 7, how many 5digit numbers can be formed that have distinct digits and are multiples of 12?
 36
 60
 84
 72
Choice B
60 
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 nonzero units digit in both base 6 and base 7 notations?
 143
 200
 157
 122
Choice A
143 
CAT Permutation and Combination: NonZero Numbers
All numbers from 1 to 150 (in decimal system) are written in base 6 notation. How many of these will contain zero's?
 25
 20
 35
 45
Choice D
45 
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?
 24
 18
 15
 12
Choice D
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?
 975
 960
 978
 986
Choice C
978 
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"?
 36
 74
 42
 86
Choice B
74 
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?
 \\frac{7}{10}\\)
 \\frac{15}{23}\\)
 \\frac{2}{3}\\)
 \\frac{11}{17}\\)
Choice B
\\frac{15}{23}\\) 
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?
 \\frac{7}{10}\\)
 \\frac{8}{11}\\)
 \\frac{1}{3}\\)
 \\frac{1}{2}\\)
Choice C
\\frac{1}{3}\\) 
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?
 6720
 2400
 4320
 1800
Choice B
2400 
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?
 7
 9
 14
 15
Choice A
7 
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?
 \\frac{1}{3}\\)
 \\frac{1}{6}\\)
 \\frac{2}{5}\\)
 \\frac{3}{8}\\)
Choice A
\\frac{1}{3}\\) 
CAT Probability: Number Theory
N is a 3digit number that is a multiple of 7; what is the probability that it will be a multiple of 5?
 \\frac{1}{5}\\)
 \\frac{11}{54}\\)
 \\frac{13}{64}\\)
 \\frac{13}{66}\\)
Choice C
\\frac{13}{64}\\) 
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?
 \\frac{2}{5}\\)
 \\frac{1}{2}\\)
 \\frac{1}{3}\\)
 \\frac{2}{3}\\)
Choice D
\\frac{2}{3}\\) 
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?

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?
 52728
 405646
 685464
 4056
Choice C
685464 
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?
 15
 48
 30
 60
Choice A
15 
CAT Permutation and Combination: 3 Digit Number
Find 3 digit numbers such that product of their digits is a natural number less than 5?
 11
 15
 13
 17
Choice C
13 
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?
 18
 20
 19
 17
Choice B
20 
CAT Permutation and Combination: Sum of Rearrangements
What is sum of all rearrangements of the 4digit number 3214?
 66660
 55554
 60048
 65024
Choice A
66660 
CAT Permutation and Combination: Sum of Rearrangements
What is sum of all rearrangements of the 4digit number 3321?
 29999
 27777
 28888
 29997
Choice D
29997 
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?
 ^{22}C_{3}
 ^{22}C_{3}  (^{8}C_{3}+ ^{7}C_{3} +^{10}C_{3})
 ^{22}C_{3} + (^{8}C_{3}+ ^{7}C_{3} +^{10}C_{3})
 ^{8}C_{3}+ ^{7}C_{3} +^{10}C_{3}
Choice B
^{22}C_{3}  (^{8}C_{3}+ ^{7}C_{3} +^{10}C_{3} 
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?
 \\frac{39}{70}\\)
 \\frac{41}{70}\\)
 \\frac{29}{70}\\)
 \\frac{17}{35}\\)
Choice D
\\frac{17}{35}\\) 
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?
 3 ! * ^{5}C_{3}
 4 ! * ^{4}C_{3}
 3 * ^{4}C_{3}
 3 ! * ^{4}C_{3}
Choice C
3 * ^{4}C_{3} 
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?
 5 ! ways
 7 ! ways
 8 ! ways
 4 ! ways
Choice A
5 ! ways 
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?
 6 * 5 ! ways
 7 * 5 ! ways
 3 * 3 ! ways
 15 * 5 ! ways
Choice D
15 * 5 ! ways 
CAT Permutation and Combination: 3 Digit numbers > 500
How many 3digit numbers greater than 500 contain the digit 9 appearing at least once?
 191
 176
 153
 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.

CAT 2023 Slot 1  QA
The number of all natural numbers up to 1000 with nonrepeating digits is
 585
 504
 648
 738
Choice D
738 
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
 1440
 1200
 1420
 1480
Choice A
1440 
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

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

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% 
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% 
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

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

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

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

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 leftmost position? [TITA]
50

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
 101
 99
 87
 105
Choice B
99 
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

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?
 16
 20
 14
 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.
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?
 0.91
 0.87
 0.94
 0.73
 0.8
Choice B
0.87XAT 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?
 It is exactly 3
 It is either 3 or 4
 It is exactly 2
 It is either 2 or 3
 It is either 0 or 1
Choice A
It is exactly 3XAT 2019 Question Paper  QADI
A bag contains marbles of three coloursred, 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"?
 Statement 1 alone is suﬃcient, but statement 2 alone is not suﬃcient to answer the question.
 Statement 2 alone is suﬃcient, but statement 1 alone is not suﬃcient to answer the question.
 Statements 1 and 2 together are not suﬃcient, and additional data is needed to answer the question.
 Each statement alone is suﬃcient to answer the question.
 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.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.
 0.91
 0.5
 0.49
 0.36
 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.

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?
 \\frac{10}{21}\\)
 \\frac{1}{2}\\)
 \\frac{3}{4}\\)
 \\frac{2}{35}\\)
Choice D
\\frac{2}{35}\\)  \\frac{10}{21}\\)

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?
 24
 25
 27
 26
Choice B
25 
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?
 1988
 1976
 1966
 1956
Choice D
1956 
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?
 3388
 3376
 3366
 3327
Choice D
3327 
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?
 4845
 4800
 4855
 4955
Choice A
4845 
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?
 60
 40
 64
 82
Choice C
64 
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?
 330
 730
 720
 820
Choice C
720 
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?
 60
 72
 56
 75
Choice C
56 
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

IPMAT 2020 Question Paper  IPM Indore Quants
The probability that a randomly chosen factor of 10^{19} is a multiple of 10^{15} is
 \\frac{1}{25})
 \\frac{1}{12})
 \\frac{1}{20})
 \\frac{1}{16})
Choice C
\\frac{1}{16}) 
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
 \\frac{4}{9})
 \\frac{5}{9})
 \\frac{4}{15})
 \\frac{2}{15})
None of these!
\\frac{4}{5}) 
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 n^{th} turn, then
 f(2) = \\frac{1}{13}\\) > f(3)
 \\frac{1}{13}\\) > f(2) > f(3)
 f(3) > f(2) = \\frac{1}{13}\\)
 f(2) = f(3) = \\frac{1}{13}\\)
Choice D
f(2) = f(3) = \\frac{1}{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
 \\frac{1}{6}\\)
 \\frac{1}{4}\\)
 \\frac{1}{5}\\)
 \\frac{1}{10}\\)
Choice C
\\frac{1}{5}\\) 
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
 4
 4.667
 4.333
 3.667
Choice B
4.667
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CAT Preparation Online  CAT Permutation and Combination; Probability Videos On YouTube
1. Digit Numbers  CAT Permutation and Combination  CAT Coaching Online
2. Numbers with Distinct Digits  CAT Permutation and Combination  CAT Coaching Online
3. Alphabetical Order  CAT Permutation and Combination  CAT Coaching Online
4. Geometric Progression  CAT Permutation and Combination  CAT Coaching Online
5. Bayes Theorem  CAT Permutation and Combination  CAT Coaching Online
6. Rearranging Letters  CAT Permutation and Combination  CAT Coaching Online
7. Possible Integer Solutions  CAT Permutation and Combination  CAT Coaching Online
8. Rearrangement of Letters  CAT Permutation and Combination  CAT Coaching Online
9. Number Theory  CAT Permutation and Combination  CAT Coaching Online
10. Distribution  CAT Permutation and Combination  CAT Coaching Online
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