I'm just unsure if I'm interpreting this problem right:
At a 12-week conference in mathematics, Sharon met seven of her friends from college. During the conference she met each friend at lunch 35 times, every pair of them 16 times, every trio eight times, every foursome four times, each set of five twice, and each set of six one, but never all seven at once. If she had lunch every day during the 84 days of the conference, did she ever have lunch alone?
So I've been working this out in my head for a while and I'm fairly sure the following is correct, but can anyone verify? $84 - 35 \binom{7}{1} + 16 \binom{7}{2} - 8 \binom{7}{3} + 4 \binom{7}{4} - 2 \binom{7}{5} + \binom{7}{6}=0$
And this is basically taking the number of days of her conference and subtracting all the ones that she had lunch with people. Because we have a result of zero, she didn't have lunch alone at all.. Correct?