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HMMT 二月 2002 · ADV 赛 · 第 3 题

HMMT February 2002 — ADV Round — Problem 3

专题
Discrete Math / 离散数学
难度
L3
来源
HMMT

题目详情

  1. How many four-digit numbers are there in which at least one digit occurs more than once?
解析
  1. How many four-digit numbers are there in which at least one digit occurs more than once? Solution: 4464 . There are 9000 four-digit numbers altogether. If we consider how many four-digit numbers have all their digits distinct, there are 9 choices for the first digit (since we exclude leading zeroes), and then 9 remaining choices for the second digit, then 8 for the third, and 7 for the fourth, for a total of 9 · 9 · 8 · 7 = 4536. Thus the remaining 9000 − 4536 = 4464 numbers have a repeated digit.