HMMT 二月 2000 · GEN 赛 · 第 18 题

HMMT February 2000 — GEN Round — Problem 18

Y ou use a lo k with four dials, ea h of whi h is set to a n um b er b et w een 0 and 9 (in lusiv e). Y ou an nev er remem b er y our o de, so normally y ou just lea v e the lo k with ea h dial one higher than the orre t v alue. Unfortunately , last nigh t someone hanged all the v alues to 5. All y ou remem b er ab out y our o de is that none of the digits are prime, 0, or 1, and that the a v erage v alue of the digits is 5. Ho w man y om binations will y ou ha v e to try?

解析

The only digits p ossible are 4, 6, 8, and 9. The only groups of n um b ers allo w ed k eeping the a v erage at 5 are 8444 are 6464. There are 4 w a ys to arrange 8444 and 6 w a ys to arrange 6464 so there are only 10 om binations to try .