A woman has 11 close Friend. Find the number of ways she can invite 5 of them to dinner if

Question

a. but where 2 of the friends are married and will not attend separately
b. where 2 of the friend will not attend together

the answer to a. 210 b. 252

Please explain....

Answer 1

Denote n choose k by nCk. nCk = n!/(k!(n-k)!)

a.
9 of the friends can attend separately
2 must attend together

There are 9C5 ways to not invite the married couple.

There are 9C3 ways to invite the married couple and 3 other friends.

Total: 9C5 + 9C3

b.
9 of the friends will attend without restriction
2 friends cannot attend together

There are 9C5 ways to invite 5 of the group of 9 friends.

There are (2C1)(9C4) ways to invite 4 of the group of 9 friends, and 1 of the other group.

Total: 9C5 + (2C1)(9C4)

Answer 2

Hmm...the book says it's 252, which corresponds to (2C1)(9C4) or 252.

Answer 3

There are 2 friends who won't come together. (2C1)(9C4) only counts the times in which one of those 2 friends are invited. There are 9C5 ways to invite neither of those 2 friends.