PUMaC 2021 · 团队赛 · 第 4 题

PUMaC 2021 — Team Round — Problem 4

Abby and Ben have a little brother Carl who wants candy. Abby has 7 different pieces of candy and Ben has 15 different pieces of candy. Abby and Ben then decide to give Carl some candy. As Ben wants to be a better sibling than Abby, so he decides to give two more pieces of candy to Carl than Abby does. Let N be the number of ways Abby and Ben can give Carl candy. Compute the number of positive divisors of N . P ∞ | n | n

解析

Abby and Ben have a little brother Carl who wants candy. Abby has 7 different pieces of candy and Ben has 15 different pieces of candy. Abby and Ben then decide to give Carl some 1 candy. As Ben wants to be a better sibling than Abby, so he decides to give two more pieces of candy to Carl than Abby does. Let N be the number of ways Abby and Ben can give Carl candy. Compute the number of positive divisors of N . Proposed by: Nancy Xu Answer: 96 If Abby gives Carl n pieces of candy, then Ben gives Carl n + 2 pieces of candy, so for a fix n 7 15 there are · ways of giving candy, where 0 ≤ n ≤ 7. Then we have: n n +2 7 7 13 X X X 7 15 7 15 7 15 · = · = · . n n + 2 n 13 − n n 13 − n n =0 n =0 n =0 P 13 7 15 22 22 2 Applying Vandermonde’s identity gives us · = , where = 2 · 5 · 7 · n =0 n 13 − n 13 13 5 11 · 17 · 19 and has 3 · 2 = 96 divisors. P ∞ | n | n