HMMT 二月 2007 · COMB 赛 · 第 2 题

英文解析

2. [ 3 ] How many 5-digit numbers abcde exist such that digits b and d are each the sum of the digits totheir immediate left and right? (That is, b = a + c and d = c + e .)
   Answer: 330 . Note that a > 0, so that b > c , and e ≥ 0 so that d ≥ c . Conversely, for each choiceof ( b, c, d ) with b > c and d ≥ c , there exists a unique pair ( a, e ) such that abcde is a number havingthe desired property. Thus, we compute
   9 9
   ∑ ∑
   (9 − c )(10 − c ) = c − 19 c + 90 = 330 .2
   c =0 c =0