Question

A woman has $11$ close friends. Find the number of ways in which she can invite $5$ them to dinner if two particular of them are not on speaking terms and will not attend together

Answer 1

Given that a woman has 11 close friends .She invites 5 of them to dinner.

Case 1: When 1 do not attend the party $\Rightarrow$ 1. ${}^9{C}_4{=}^9{C}_4$

Case 2: When 2 do not attend the party $\Rightarrow$${}^9{C}_4{=}^9{C}_4$

Case 3: When both of them do not attend the party =${}^9{C}_5$

$\therefore$ Total no: of ways =${}^9{C}_4{+}^9{C}_4{+}^9{C}_5$.

=378 ways