# CAT Probability, Combinatorics Questions With Video Solutions

Probability topic an important part of the CAT. You can expect close to 2-3 questions in the latest 22 question format of the CAT Quant section. Here, some of the important probability questions for the CAT Exam. If you want to practice these important probability questions, you can download the PDF, which is completely Free. Probability is often one of the most feared topics among the candidates. It is not a very difficult topic if you understand the basics of Probability well. Practising free CAT mocks where you'll get a fair idea of how questions are asked, and type of questions asked in CAT Exam. Probability-based questions appear in the CAT test almost every year. A lot of aspirants avoid this topic but remember that one can definitely solve the easy questions on Probability if one is thorough with the basics. Therefore, practising questions with Probability should not be avoided. The chances of occurring or not occurring an event should be determined based on the number of favourable and not favourable conditions.

Here we are giving some very important probability questions, which also include questions from the CAT previous papers. The candidates are advised to try each question on their own and later go through the solutions given below. Note:

## Topic Weightage Over Past 6 years

 Year Weightage 2023 0 2022 3 2021 4 2020 0 2019 1 2018 3

## CAT Probability And Combinatorics Formulas PDF

CAT Probability and Combinatorics are among the important topics in the quantitative aptitude section, and it is vital to understand the formulas related to them clearly. To help the aspirants to ace this topic, we have made a PDF containing a comprehensive list of formulas, tips, and tricks that you can use to solve Progressions and series questions with ease and speed. Click on the below link to download the CAT Progressions and series formulas PDF.

1. Basics of Probability

If the probability of an event occurring is p, then the probability that the event will occur r times in n trials is given by = $$^{n}\textrm{C}_{r}p^{r}(1-p)^{n-r}$$

If p is the probability of an event, then odds in favor of an event are p / (1 – p). Conversely, the odds against are (1-p)/p.

Say $$E_{1}, E_{2}…. E_{n}$$ are mutually exclusive exhaustive events with probabilities $$p_{1}, p_{2}.... p_{n}$$ and expected values $$e_{1}, e_{2}.... e_{n}$$ then Expected payoff = $$\sum_{i=1}$$ $$^{n}p_{i}e_{i}$$

2.  Bayes theorem:

Let $$E_{1}, E_{2}, E_{3}...$$ be mutually exclusive and collectively exhaustive events each with a probability $$p_{1}, p_{2}, p_{3}...$$ of occurring. Let B be another event of non-zero probability such that probability of B given $$E_{1}$$ is $$q_{1}$$, B given $$E_{2}$$ is $$q_{2}$$ etc. By Bayes theorem: $$P(E_{i}/B) = \frac{p_{i}q_{i}}{\sum_{j=1}^{n}p_{j}q_{j}}$$\$

3. Derangements

If n distinct items are arranged, the number of ways they can be arranged so that they do not occupy their intended spot is $$D = n!$$($$\frac{1}{0!}$$ - $$\frac{1}{1!}$$ + $$\frac{1}{2!}$$ - $$\frac{1}{3!}$$ + .... + $$\frac{(-1)^{n}}{n!}$$)

For, example, Derangements of 4 will be D(4) = $$4!\left(1-\dfrac{1}{1!}+\dfrac{1}{2!}-\dfrac{1}{3!}+\dfrac{1}{4!}\right)=24\left(\dfrac{1}{2}-\dfrac{1}{6}+\dfrac{1}{24}\right)=24\left(\dfrac{12-4+1}{24}\right)=9$$

D(1) = 0, D(2) = 1, D(3) = 2, D(4) = 9, D(5) = 44, and D(6) = 265

4. Arrangement with repetitions

If x items out of n items are repeated, then the number of ways of arranging these n items is $$\dfrac{n!}{x!}$$ ways. If x items, y items and z items are repeated within n items, they can be arranged in $$\frac{n!}{a!b!c!}$$ ways.

CAT probability and combinatorics questions are the important questions that frequently appear in the CAT examination. These questions require a solid understanding of fundamental concepts such as permutations, combinations, and probability distributions. As such, CAT aspirants need to grasp these topics to excel in the exam thoroughly. To help the aspirants, we have compiled all the questions from this topic that appear in the previous CAT papers, along with the video solutions for every question explained in detail by the CAT experts. One can download them in a PDF format or take them in a test format. Click on the link below to download the CAT probability and combinatorics questions with detailed video solutions PDF.

## CAT 2022 Probability, Combinatorics questions

#### Question 1

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

#### Question 2

The arithmetic mean of all the distinct numbers that can be obtained by rearranging the digits in 1421, including itself, is

#### Question 3

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

## CAT 2021 Probability, Combinatorics questions

#### Question 1

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

#### Question 2

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

#### Question 3

How many three-digit numbers are greater than 100 and increase by 198 when the three digits are arranged in the reverse order?

#### Question 4

A four-digit number is formed by using only the digits 1, 2 and 3 such that both 2 and 3 appear at least once. The number of all such four-digit numbers is

## CAT 2019 Probability, Combinatorics questions

#### Question 1

With rectangular axes of coordinates, the number of paths from (1, 1) to (8, 10) via (4, 6), where each step from any point (x, y) is either to (x, y+1) or to (x+1, y), is

## CAT 2018 Probability, Combinatorics questions

#### Question 1

How many two-digit numbers, with a non-zero digit in the units place, are there which are more than thrice the number formed by interchanging the positions of its digits?

#### Question 2

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

#### Question 3

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

## CAT 2017 Probability, Combinatorics questions

#### Question 1

The numbers 1, 2, ..., 9 are arranged in a 3 X 3 square grid in such a way that each number occurs once and the entries along each column, each row, and each of the two diagonals add up to the same value.

If the top left and the top right entries of the grid are 6 and 2, respectively, then the bottom middle entry is

#### Question 2

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?

#### Question 3

Let AB, CD, EF, GH, and JK be five diameters of a circle with center at 0. 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?

#### Question 4

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?

#### Question 5

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?

## CAT 2008 Probability, Combinatorics questions

#### Question 1

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?

#### Question 2

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

Instruction for set 1:

Directions for the next two questions: The figure below shows the plan of a town. The streets are at right angles to each other. A rectangular park (P) is situated inside the town with a diagonal road running through it. There is also a prohibited region (D) in the town.

#### Question 3

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 can choose is

[CAT 2008]

Instruction for set 1:

Directions for the next two questions: The figure below shows the plan of a town. The streets are at right angles to each other. A rectangular park (P) is situated inside the town with a diagonal road running through it. There is also a prohibited region (D) in the town.

#### Question 4

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 can choose is

[CAT 2008]