The total number of ways in which the woman can invite 5 out of her 11 close friends to dinner, ensuring that the two particular friends who are not on speaking terms do not attend together, is 378.
How to find the number of ways ?
Neither of the two friends attends: This is the easiest case. We simply choose 5 friends out of the remaining 9 excluding the two who aren't on speaking terms. This can be done in:
9C5 = 126 ways
One of the two friends attends: In this case, we can choose the remaining 4 friends from the 9 in:
= 9C4
= 126 ways
Additionally, we need to choose which of the two feuding friends attends, which can be done in 2 ways. This gives us a total of:
9C4 x 2 = 252 ways
Therefore, the total number of ways is the sum of these two cases:
= 126 + 252
= 378