Tiffany has 14 classmates; all of her classmates have a different number of friends in the class. How many of them are friends with Tiffany? (If A is a friend of B, then B is a friend of A.)
There are 15 people in the class, and the number of friends can only go from 0 to 14, and if there is someone with 0 friends, there will be no one with 14 friends. So at most there are 14 distinct number of friends, and it is either 0-13 or 1-14. And note that if Tiffany's 14 classmates have 0-13 number friends, we can change that to 1-14 simply by making the one with 0 friend to be friends with those who had 7-13 friends (making those to have 8-14 friends, and the one who had 0 friend to have 7 friends, not changing those with 1-6 friends, so we have 1-14). Since this doesn't affect the number of friends Tiffany has, we can assume that it is the case of 1-14 number of friends that is happening here (the same reasoning below can work with 0-13).
Let the 14 friends be called C1, C2, ..., C14, according to the number of friends they have. So C1 has 1 friend, C2 has 2 friends, and so on. So we have Tiffany + C1-C14 (should we call it "Tiffany and the 14 C's"?)
Now notice that:
- The one with 14 friends (C14) should be friends with everyone else, including Tiffany. Now this makes the one with only one friend (C1) not friend with Tiffany, since C1 is already friend with C14 and C1 only has one friend. From here we know that Tiffany is friend with C14, and not with C1.
- The one with 13 friends (C13) should be friends with everyone else except C1. Now this makes the one with two friends (C2) not friend with Tiffany, since C2 is already friends with C14 and C13. From here we know that Tiffany is friend with C13, and not with C2.
- Continuing the pattern of pairing off C(N) with C(15-N), we have C14, C13, C12, C11, C10, C9, and C8 as Tiffany's friends. And the rest (C1, C2, ..., C7) are not friends with Tiffany.
Tiffany has 7 friends.