HMMT 二月 2017 · COMB 赛 · 第 2 题

HMMT February 2017 — COMB Round — Problem 2

How many ways are there to insert +’s between the digits of 111111111111111 (fifteen 1’s) so that the result will be a multiple of 30?

解析

How many ways are there to insert +’s between the digits of 111111111111111 (fifteen 1’s) so that the result will be a multiple of 30? Proposed by: Yang Liu Answer: 2002 Note that because there are 15 1’s, no matter how we insert +’s, the result will always be a multiple of 3. Therefore, it suffices to consider adding +’s to get a multiple of 10. By looking at the units digit, we need the number of summands to be a multiple of 10. Because there are only 15 digits in our number, we have to have exactly 10 summands. Therefore, we need to insert 9 +’s in 14 possible ( ) 14 positions, giving an answer of = 2002 . 9