# Questionbank: Permutation and Probability

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This section hosts a number of CAT Level questions on Permutation and Combination, and Probability. For a free PDF download of the Permutation and Combination chapter authored by Mr. Rajesh, click here.
1. ### 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
2. ### 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
3. ### 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
4. ### Permutation 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
5. ### 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
11
• Counting - Number Systems
• Medium
6. ### 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
7. ### Combinatorics 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
3980
• Counting - Number Systems
• Medium
8. ### Number Systems - Combinatorics

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
60
• Counting - Number Systems
• Medium
9. ### 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
10. ### 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
11. ### 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
12
• Counting - Number Systems
• Medium
12. ### 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
13. ### 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
14. ### Probability and 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. ${7 \over 10}$
2. ${15 \over 23}$
3. ${2 \over 3}$
4. ${11 \over 17}$
15. ### 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. ${7 \over 10}$
2. ${8 \over 11}$
3. ${1 \over 3}$
4. ${1 \over 2}$
16. ### 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
17. ### 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
18. ### Rearrangement of Letters - Probability

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

1. 1/3
2. 1/6
3. 2/5
4. 3/8
19. ### Probability and 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. 1/5
2. 11/54
3. 13/64
4. 13/66
20. ### 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. 2/5
2. 1/2
3. 1/3
4. 2/3
21. ### 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?

1. 720
2. 480
3. 360
4. 350
22. ### 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
23. ### Rolling a Die

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

1. 30
2. 48
3. 54
4. 60
24. ### 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
25. ### 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
• 3 Digit Number - Sum of Digits
• Medium
26. ### Sum of Rearrangements

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

1. 66660
2. 55554
3. 60048
4. 65024
27. ### Sum of Rearrangements

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

1. 29999
2. 27777
3. 28888
4. 29997
28. ### 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
29. ### Selection and Probability

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. ${\frac{17}{35}}$

30. ### 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 ! * $^{5}C_{3}$
2. 4 ! * $^{4}C_{3}$
3. 3 * $^{4}C_{3}$
4. 3 ! * $^{4}C_{3}$
31. ### 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
32. ### 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
33. ### 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

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