A woman has 11 close friends. Find the number of ways she can invite 5 of them to dinner where: (a) there are no restrictions. (b) Two of the friends are married to each other and will not attend separately. (c) Two of the friends are not speaking with each other and will not attend together
(a) There are no restrictions, so we can simply use the formula for combinations:
${11 \choose 5} = \frac{11!}{5!6!} = 462$
Therefore, there are 462 ways for the woman to invite 5 of her 11 close friends to dinner.
(b) Two of the friends are married and will not attend separately, so we first need to choose one of the couples to invite. There are ${11 \choose 2}$ ways to do this. Once we have chosen the couple, we only need to choose 3 more friends from the remaining 9. There are ${9 \choose 3}$ ways to do this. Therefore, the total number of ways the woman can invite 5 friends to dinner, taking into account the married couple, is:
${11 \choose 2} \cdot {9 \choose 3} = \frac{11!}{2!9!} \cdot \frac{9!}{3!6!} = 3300$
Therefore, there are 3300 ways for the woman to invite 5 of her 11 close friends to dinner, taking into account the married couple.
(c) Two of the friends are not speaking with each other and will not attend together. We can solve this problem by counting the number of ways that the two friends who are not speaking will be invited together and then subtracting this from the total number of ways without restrictions. There are ${2 \choose 2}$ ways to choose both of the non-speaking friends, and ${9 \choose 3}$ ways to choose the remaining 3 friends from the other 9. There are therefore:
${2 \choose 2} \cdot {9 \choose 3} = {9 \choose 3} = \frac{9!}{3!6!} = 84$
ways to invite the 5 friends to dinner with the non-speaking friends. Therefore, the number of ways to invite the 5 friends to dinner without the non-speaking friends is:
${11 \choose 5} - {2 \choose 2} \cdot {9 \choose 3} = 462 - 84 = 378$
Therefore, there are 378 ways for the woman to invite 5 of her 11 close friends to dinner, without inviting the non-speaking friends together.