The CAT QA section requires speed and accuracy, along with a thorough understanding of the Combinatorics. This article provides a set of MCQs on Combinatorics to help you understand the topic and improve your problem-solving skills with the help of detailed solutions by ensuring conceptual clarity, which will help you in the CAT 2025 exam preparation
Whether you're revising the basics or testing your knowledge, these MCQs will serve as a valuable practice resource.
The CAT 2025 exam is expected to follow a similar trend to the CAT 2024, with 22 questions from the QA section out of a total of 68 questions.
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CAT MCQs on Combinatorics
1. A man has 9 friends: 4 boys and 5 girls. In how many ways can he invite them, if there have to be exactly 3 girls in the invitees?
A
\(\binom{5}{3} \times \binom{4}{2}\)
B
\(\binom{5}{3} \times \binom{4}{3}\)
C
\(\binom{5}{3} \times \binom{4}{4}\)
D
\(\binom{5}{3} \times \binom{4}{1}\)
2. 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 two 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 lines are required?
A
72
B
90
C
96
D
144
3. Find the total number of ways in which one can wear three distinct rings on the five fingers of one’s right hand, given that one is allowed to wear more than one ring on a finger.
A
120
B
360
C
480
D
210
4. A is a non-empty set having n elements. P and Q are two subsets of A such that P ⊆ Q. Find the number of ways of choosing the subsets P and Q.
A
4n
B
3n
C
2n
D
n2
5. In how many ways can 40 sweets be given to A, B, C, and D such that:
- B gets at least 3 sweets,
- D gets at least 5 sweets,
- A and C may get zero sweets.
- B gets at least 3 sweets,
- D gets at least 5 sweets,
- A and C may get zero sweets.
A
4960
B
6545
C
9139
D
15
6. Neelam rides her bicycle from her house at \(A\) to her office at \(B\), taking the shortest path. The number of possible shortest paths that she can choose is:
A
60
B
75
C
45
D
90
7. Neelam rides her bicycle from her house at \(A\) to her club at \(C\), via \(B\) taking the shortest path. The number of possible shortest paths that she can choose is:
A
1170
B
630
C
792
D
1200
8. In a chess competition with boys and girls, each student plays exactly one game with each other student. Total 45 games were boy-boy, 190 games boy-girl. Number of girls = ?
A
200
B
216
C
235
D
256
9. Suppose you have a currency, named Miso, in three denominations: 1 Miso, 10 Misos and 50 Misos. In how many ways can you pay a bill of 107 Misos?
A
17
B
16
C
18
D
15
10. In a tournament, there are \( n \) teams \( T_1, T_2, \ldots, T_n \), with \( n > 5 \). Each team consists of \( k \) players, \( k > 3 \). The following pairs of teams have one player in common: \( T_1 \) & \( T_2 \), \( T_2 \) & \( T_3 \), \( \ldots \), \( T_{n-1} \) & \( T_n \), \( T_n \) & \( T_1 \). No other pair of teams has any player in common. How many players are participating in the tournament, considering all the \( n \) teams together?
A
\( n(k-1) \)
B
\( k(n-1) \)
C
\( n(k-2) \)
D
\( k(n-2) \)
11. 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 two 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 lines are required?
A
72
B
90
C
96
D
144
12. 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-minute song one pair after the other. If the total time taken for singing is 28 minutes, what is \( N \)?
A
5
B
7
C
9
D
None of the above
13. In the adjoining figure, the lines represent one-way roads allowing travel only northwards or only westwards. Along how many distinct routes can a car reach point B from point A?
A
15
B
56
C
120
D
336
14. 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 be done so that no two adjacent stripes have the same colour is
A
\( 12 \times 81 \)
B
\( 16 \times 192 \)
C
\( 20 \times 125 \)
D
\( 24 \times 216 \)
15. If there are 10 positive real numbers \( n_1 < n_2 < n_3 \dots < n_{10} \), how many triplets of these numbers \( (n_1, n_2, n_3), (n_2, n_3, n_4), \dots \) can be generated such that in each triplet the first number is always less than the second number, and the second number is always less than the third number?
A
45
B
90
C
120
D
180
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