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Q.

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 ________.

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a

C 8 5

b

2 × C 8 5

c

C 10 5 − 2 × C 8 5

d

None of these

answer is B.

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Detailed Solution

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

C 8 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

C 8 5

.
Hence, the total number of ways in which the friends can be invited to the party is

C 8 3 + C 8 5

.
By using the property of combination

C n r = C n n − r

, we have

C 8 5 = C 8 3

.
So, the total number of ways to invite 5 friends is

2 × C 8 5

.
Hence, the correct option is 2.

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