Question

Kanchan has 10 friends among whom two are married to each other. She wishes to invite five of them for a party. If the married couple refuses to attend separately, then the number of different ways in which she can invite five friends is
(a) $ {}^{8}{{C}_{5}} $
(b) $ 2\times {}^{8}{{C}_{5}} $
(c) $ {}^{10}{{C}_{5}}-2\times {}^{8}{{C}_{4}} $
(d) none of these

Answer 1

Hint: We will consider two possibilities for attendance at the party. Then we will find the number of ways to invite five people considering the first possibility. After that, we will consider the second possibility and find the number of ways in which five people can be invited in this case. Since both the possibilities cannot occur simultaneously, we will add the number of ways in both possibilities to find the total number of ways.

Complete step by step answer:
Kanchan has 10 friends and 2 of them are married to each other. Kanchan has to invite 5 people to the party. Now, since the married couple refuses to attend separately, we have two possibilities.
The first possibility is that the married couple is invited. Now, we are left with 3 more invitations and 8 friends. So, we will choose 3 friends from 8. The number of ways in which we can do this is $ {}^{8}{{C}_{3}} $ .
The second possibility is that the married couple is not invited. Now, we have 5 invitations and 8 friends. Hence, we will choose 5 friends from the remaining 8. The number of ways in which we can invite 5 friends out of 8 is $ {}^{8}{{C}_{5}} $ .
These two possibilities cannot occur simultaneously. Hence, the total number of ways in which the friends can be invited to the party is $ {}^{8}{{C}_{3}}+{}^{8}{{C}_{5}} $ .
We know the property of the combinations that if we are choosing $ r $ objects out of $ n $ , then $ {}^{n}{{C}_{r}}={}^{n}{{C}_{n-r}} $ . Therefore, we have $ {}^{8}{{C}_{5}}={}^{8}{{C}_{3}} $ . So, the total number of ways to invite 5 friends is $ 2\times {}^{8}{{C}_{5}} $ and the correct option is (b).

Note:
It is useful to know the properties of permutations and combinations. Here, we can treat the married couple as one object while deciding whether or not Kanchan should invite them. Hence, we come up with two possibilities. If we are finding the number of ways to do two different things simultaneously, then we multiply the number of ways of doing each thing separately. This is also called the multiplication principle of counting.