CAT Permutation, Combination & Probability questions (1991–2025) — actual Quant questions with solutions

Breakup G Strategy | PnC + Factors + Maxima Minima | HARD | CAT 2025 Slot 1

In a 3-digit number N, the digits are non-zero and distinct such that none of the digits is a square, and only one of the digits is a prime number. Then, the number of factors of the minimum possible value of N is ______

PnC G Strategy | Concepts of PnC | EASY | CAT 2025 Slot 1

A cafeteria offers 5 types of sandwiches. Moreover, for each type of sandwich, a customer can choose one of 4 breads and opt for either small or large sized sandwich. Optionally, the customer may also add up to 2 out of 6 available sauces. The number of different ways in which an order can be placed for a sandwich, is

PnC Basics

Find the number of all positive integers up to 500 with non-repeating digits.

P, Q, R, and S are towns. Paths between them are defined. 62 ways exist from P to S; 27 ways from Q to R. Find number of direct paths between Q and R.

Number of all positive integers ≤ 500 with non-repeating digits:

Neelam rides her bicycle from her house at A to her office at B, taking the shortest path. Then the number of possible shortest paths that she ca choose is

Neelam rides her bicycle from her house at A to her club at C, via B taking the shortest path. Then the number of possible shortest paths that she ca choose is

How many integers, greater than 999 but not greater than 4000, can be formed with the digits 0, 1, 2, 3 and 4, if repetition of digits is allowed?

What is the number of distinct terms in the expansion of (a + b + c)20?

In a tournament, there are n teams T1 , T2 , Tn with n > 5. Each team consists of k players, k > 3. The following pairs of teams have one player i common: T1 & T2 , T2 & T3 ,……, Tn − 1 & Tn , and Tn & T1. No other pair of teams has any player in common. How many players are participating in the tournament, considering all the n teams together?

There are 6 tasks and 6 persons. Task 1 cannot be assigned either to person 1 or to person 2; task 2 must be assigned to either person 3 or person 4 Every person is to be assigned one task. In how many ways can the assignment be done?

In a chess competition involving some boys and girls of a school, every student had to play exactly one game with every other student. It was foun that in 45 games both the players were girls, and in 190 games both were boys. The number of games in which one player was a boy and the othe was a girl is

Let S be the set of five digit numbers formed by the digits 1, 2, 3, 4 and 5, using each digit exactly once such that exactly two odd positions ar occupied by odd digits. What is the sum of the digits in the rightmost position of the numbers in S?

N persons stand on the circumference of a circle at distinct points. Each possible pair of persons, not standing next to each other, sings a two-minut song one pair after the other. If the total time taken for singing is 28 minutes, what is N?

In the adjoining figure, the lines represent one-way roads allowing travel only northwards or only westwards. Along how many distinct routes can a ca reach point B from point A?

A new flag is to be designed with six vertical stripes using some or all of the colours yellow, green, blue and red. Then, the number of ways this can b done such that no two adjacent stripes have the same colour is:

Each question is followed by two statements, A and B. Answer each question using the following instructions Choose 1 if the question can be answered by using one of the statements alone but not by using the other statement alone. Choose 2 if the question can be answered by using either of the statements alone. Choose 3 if the question can be answered by using both statements together but not by either statement alone. Choose 4 if the question cannot be answered on the basis of the two statements. Tarak is standing 2 steps to the left of a red mark and 3 steps to the right of a blue mark. He tosses a coin. If it comes up heads, he moves one step t the right; otherwise he moves one step to the left. He keeps doing this until he reaches one of the two marks, and then he stops. At which mark doe he stop? He stops after 21 coin tosses. He obtains three more tails than heads.

Twenty-seven persons attend a party. Which one of the following statements can never be true?

How many three digit positive integers, with digits x, y and z in the hundred's, ten's and unit's place respectively, exist such that x < y, z < y and x ≠ 0?

There are 6 boxes numbered 1, 2,…, 6. Each box is to be filled up either with a red or a green ball in such a way that at least 1 box contains a green ba and the boxes containing green balls are consecutively numbered. The total number of ways in which this can be done is

A graph may be defined as a set of points connected by lines called edges. Every edge connects a pair of points. Thus, a triangle is a graph with edges and 3 points. The degree of a point is the number of edges connected to it. For example, a triangle is a graph with three points of degree each. Consider a graph with 12 points. It is possible to reach any point from any other point through a sequence of edges. The number of edges, e, i the graph must satisfy the condition

There are 12 towns grouped into four zones with three towns per zone. It is intended to connect the towns with telephone lines such that every tw towns are connected with three direct lines if they belong to the same zone, and with only one direct line otherwise. How many direct telephone line are required?

An intelligence agency forms a code of two distinct digits selected from 0, 1, 2, … , 9 such that the first digit of the code is non-zero. The code handwritten on a slip, can however potentially create confusion, when read upside down – for example, the code 91 may appear as 16. How man codes are there for which no such confusion can arise?

n1, n2, n3 … n10 are 10 numbers such that n1 > 0 and the numbers are given in ascending order. How many triplets can be formed using these number such that in each triplet, the first number is less than the second number, and the second number is less than the third number?

How many numbers between 0 and one million can be formed using 0, 7 and 8?

In how many ways, we can choose a black and a white square on a chess board such that the two are not in the same row or column?

How many four-lettered passwords can be formed by using symmetrical letters only? (repetitions not allowed)

How many three-lettered words can be formed such that at least one symmetrical letter is there?

The figure below shows the network connecting cities A, B, C, D, E and F. The arrows indicate permissible direction of travel. What is the number o distinct paths from A to F?

Let n be the number of different 5 digit numbers, divisible by 4 with the digits 1, 2, 3, 4, 5 and 6, no digit being repeated in the numbers. What is th value of n?

One red flag, three white flags and two blue flags are arranged in a line such that, no two adjacent flags are of the same colour. the flags at the two ends of the line are of different colours. In how many different ways can the flags be arranged?

Sam has forgotten his friend’s seven-digit telephone number. He remembers the following: the first three digits are either 635 or 674, the number i odd, and the number nine appears once. If Sam were to use a trial and error process to reach his friend, what is the minimum number of trials he ha to make before he can be certain to succeed?

There are three cities A, B and C, each of these cities is connected with the other two cities by at least one direct road. If a traveller wants to go fro one city (origin) to another city (destination), she can do so either by traversing a road connecting the two cities directly, or by traversing two road the first connecting the origin to the third city and the second connecting the third city to the destination. In all there are 33 routes from A to (including those via C). Similarly there are 23 routes from B to C (including those via A). How many roads are there from A to C directly?

ABCDEFGH is a regular octagon. A and E are opposite vertices of the octagon. A frog starts jumping from vertex to vertex, beginning from A. From an vertex of the octagon except E, it may jump to either of the two adjacent vertices. When it reaches E, the frog stops and stays there. Let an be the number of distinct paths of exactly n jumps ending in E. Then what is the value of a2n – 1?

Ten points are marked on a straight-line and 11 points are marked on another straight-line. How many triangles can be constructed with vertices fro among the above points?

For a scholarship, at the most n candidates out of 2n + 1 can be selected. If the number of different ways of selection of at least one candidate is 63 the maximum number of candidates that can be selected for the scholarship is

If a 4 digit number is formed with digits 1, 2, 3 and 5. What is the probability that the number is divisible by 25, if repetition of digits is not allowed?

Data is provided followed by two statements – I and II – both resulting in a value, say I and II. As your answer, Type 1, if I > II. Type 2, if I < II. Type 3, if I = II. Type 4, if nothing can be said. The probability of encountering 54 Sundays in a leap year. The probability of encountering 53 Sundays in a non-leap year.

A box contains 6 red balls, 7 green balls and 5 blue balls. Each ball is of a different size. The probability that the red ball selected is the smallest red bal is

A player rolls a die and receives the same number of rupees as the number of dots on the face that turns up. What should the player pay for each roll he wants to make a profit of one rupee per throw of the die in the long run?

How many numbers can be formed from 1, 2, 3, 4, 5, without repetition, when the digit at the unit’s place must be greater than that in the ten’s place?