PUMaC 2011 · 组合（B 组） · 第 2 题

PUMaC 2011 — Combinatorics (Division B) — Problem 2

[ 3 ] Consider the sum ab + cde , where each of the letters is a distinct digit between 1 and 5. How many values are possible for this sum?

解析

Since there is no carrying involved, we can do casework based on the sum’s units digit. There are no sums which have a units digit of 0 , 1 , or 2. If it is 3, 4, 8, or 9 then we know which two digits were added; in each of these cases there are three possible values for the sum’s tens digit, after which the hundreds digit is determined. If it is 5, 6, or 7 then there are two possible pairs of added digits, and it is easily seen that in every such case there are five possible values for the sum’s tens digit. Therefore there are 4 · 3 + 3 · 5 = 27 possible values for the sum.