Question

A lady gives a dinner party for six guests. The number of ways in which they may be selected from among ten friends if two of the friends will not attend the party together is
(a) 112
(b) 140
(c) 164
(d) none of these

Answer 1

(b) 140

Suppose there are two friends, A and B, who do not attend the party together.
If both of them do not attend the party, then the number of ways of selecting 6 guests = ⁸C₆ = 28
If one of them attends the party, then the number of ways of selecting 6 guests = 2.⁸C₅ = 112
∴ Total number of ways = 112 + 28 = 140