A women has 11 close friends. Number of ways in which she can invite 5 of them to dinner, if two particular of them are not on speaking terms and will not attend together is

Total number of ways of selecting 5 from 11 friends=${ }^{11}{C}_5$

Number of ways of inviting 5 if  the two particular women are included=${ }^9{C}_3$

Required number of ways ${=}^{11}{C}_5{-}^9{C}_3$

or

Number of ways of inviting 5 if both the particular women are excluded=${ }^9{C}_5$

Number of ways of inviting one of the two particular women is included and the other is excluded=${ }^9{C}_4{+}^9{C}_4$

Required number of ways ${=}^9{C}_5{+}^9{C}_4{+}^9{C}_4=3\left({ }^9{C}_4\right)$