Permutation and Combination Questions for CAT 2026 — Selection and Grouping with Solutions
- Jul 22, 2025
- 5 min read
Updated: Apr 12
PERMUTAION AND COMBINATION Question for CAT
Q.1 16 persons shake hands with one another in a party. How many shake hands took place?
a.120
b.240
c.480
d.None of the above
Q.2 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?
a.56
b.60
c.72
d.75
Q.3 Ten students are participating in the race. In how many different ways can the first three prizes be won?
a.10C3
b.10P3
c.103
d.310
Q.4 A man tries to send invitation to 6 of his friends through 3 servants. Find the number of possible ways that the invitation card can be sent.
a.216
b.729
c.6! × 3!
d.None of the above
Q.5 The number of ways of selecting 15 teams from 15 men and 15 women, such that each team consists of a man and a woman, is:
a.D(15)
b.15!
c.30C15
d.30P15
Q.6 Five students are to be arranged on five chairs for a photograph. Three of these are girls and the rest are boys. Find the total number of arrangements, in which three girls are together.
a.36
b.120
c.12
d.None of the above
Q.7 From a group of 5 men and 10 women, a committee has to formed of 5 members in which at least 4 men are required. The committee can be formed in .................... ways.
a.50
b.15C5
c.51
d.None of the above
Q.8 5 Indian and 5 American couples meet at a party and shake hands. If no wife shakes hands with her husband and no Indian wife shakes hands with a male, then find the number of hand shakes that take place in the party?
a.120
b.240
c.480
d.None of the above
Q.9 A shop owner has an unlimited supply of 8 different types of flowers. If a customer asks for a five flower arrangement such that the arrangement should start and end with the same type of flower, then in how many ways the arrangement can be made by the shop owner?
a.8C4 x 3!
b.83
c.84
d.8C4 x 4C1 x 3!
Q.10 In how many ways can be select 5 cards from a card pack such that all 4 suits appear?
a.52728
b.405646
c.685464
d.4056
Q.11 In a chess tournament, all participants were to play one game with the other. Two players fell ill after having played 3 games each. If total number of games played in the tournament is equal to 84, then total number of participants in the beginning was equal to:
a.14
b.15
c.16
d.18
Q.12 There is a straight table with a capacity of 9 members and they have to sit only on one side of the table. Arrange 4 boys and 5 girls such that the boy must have two girls adjacent to the boy (on either side). Find the total number of arrangements.
a.1840
b.2160
c.2880
d.6240
Q.13 The number of ways in which a mixed double game can be arranged from amongst 9 married couples, if no husband and wife play in the same game is:
a.1512
b.1514
c.3024
d.3028
Q.14 From 6 different novels and 3 different dictionaries, 4 novels and 1 dictionary are to be selected and arranged in a row on a shelf so that the dictionary is always in the middle. Then the number of such arrangements is:
a.At least 750 but less than 1000
b.At least 1000
c.Less than 500
d.At least 500 but less than 750
Q.15 A box contains 4 different pink balls and 5 different blue balls. In how many ways can you make a selection so as to take at least 1 pink ball and 1 blue ball?
a.(24 - 1) × (25 - 1)
b.(29 - 1)
c.(9! - 1)
d.None of the above
Q.16 In how many ways can a committee of 5 persons be formed from 6 gents and 4 ladies so that there is at least one lady in the committee?
a.246
b.10C5
c.10P5
d.None of the above
Q.17 There are 280 ways of forming a group of (x + 2) men and 3 women out of a total of 8 men and 5 women. Find the value of x.
a.2
b.3
c.4
d.6
Q.18 Three men have 4 coats, 5 waist coats and 6 caps. The number of ways they can wear them is:
a.15P3
b.4C3 × 5C3 × 6C3
c.180
d.4P3 × 5P3 × 6P3
Q.19 An urn contains 5 red marbles, 4 black marbles and 3 white marbles. Then the number of ways in which 4 marbles can be drawn so that atmost three of them are red is .................... .
a.420
b.455
c.490
d.None of these
Q.20 Kavita has 10 friends and she wants to invite 6 of them to a party. How many times will 3 particular friends always attend the party?
a.10C6
b.10C3
c.7C3
d.None of the above
Q.21 A double-decker bus has 5 empty seats in the up-stair and 5 empty seats in the down stair. 10 people board the bus of which 2 are old people and 3 are children. The children refuse to take seats down stair while old people insist to stay down-stair. In how many different arrangements can be 10 people take their seats in the bus?
a.144000
b.146000
c.146400
d.None of these
Q.22 The number of ways in which 3 distinct numbers in AP can be selected from 1, 2, 3, ….24 is
a.132
b.572
c.264
Frequently Asked Questions — Selection and Grouping Problems for CAT 2026
What is a combination and when do I use it?
A combination C(n,r) = n!/[r!(n-r)!] counts the number of ways to choose r items from n without regard to order. Use combinations for: selecting committee members, choosing items from a set, forming groups where arrangement doesn't matter.
How do I solve committee selection problems with constraints?
Break into cases based on the constraint. Example: committee of 4 from 5 men and 4 women with at least 2 women. Cases: exactly 2 women (C(4,2)xC(5,2)) + exactly 3 women (C(4,3)xC(5,1)) + all 4 women (C(4,4)xC(5,0)). Sum the cases.
What is the formula for distributing n distinct objects into r distinct groups?
Each object can go into any of r groups, so total = r^n. If groups must be non-empty (surjective): use inclusion-exclusion = sum over k of (-1)^k x C(r,k) x (r-k)^n. For equal-sized groups, divide by the group count factorial if groups are identical.
How do I solve 'at least one from each group' selection problems?
Use inclusion-exclusion or complementary counting. For selecting at least 1 from group A (size a) and at least 1 from group B (size b) to form a committee of r: Total C(a+b,r) - C(a,r) [none from B] - C(b,r) [none from A].
What is the concept of identical vs distinct in grouping?
Identical groups: divide by k! if forming k identical groups (since group labels don't matter). Distinct groups: no division needed (group labels matter). This distinction critically affects the answer.
How do I count selections from mixed repeated items?
For selecting r items from groups of identical items (e.g., 3 identical red, 4 identical blue, 5 identical green): use generating functions or case analysis. The generating function for each group is (1 + x + x^2 + ... + x^k) where k is the group size.
What is the multinomial coefficient and when is it used?
The multinomial coefficient n!/(n1! x n2! x ... x nk!) counts arrangements of n items where n1 are of type 1, n2 of type 2, etc. It generalises both permutations with repetition and combinations. Used in dividing groups of identical/distinct objects.
How should I approach selection problems in CAT?
First identify: distinct or identical? Ordered or unordered? With or without repetition? Then apply the appropriate formula. For complex problems, break into exhaustive non-overlapping cases and sum. Practice 10 varied selection problems per topic.
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