AMC 8 2011 Problem 23
AMC8 2011 年第 23 题
排列与组合★★★☆☆
How many 4-digit positive integers have four different digits, where the leading digit is not zero, the integer is a multiple of 5, and 5 is the largest digit?
有多少个各个位上数字不同的 4 位数，满足以下条件：最高位不为 0，这个 4 位数是 5 的倍数，且 5 是最大的数字？
- A.24
- B.48
- C.60
- D.84
- E.108
正确答案 · Correct Answer
D
解析 · Solution
Solution 1
Any integer that is a multiple of 5 5 must have a 0 0 or 5 5 as the units digit. If the units digit is 0 0, then the other three digits must be a 5 5 and two digits selected from 1, 2, 3 1,2,3, and 4 4. After the pair is selected, there are 6 6 possible ways to arrange them to form the number. So there are 6 ⋅ 6 = 36 6⋅6=36 possible numbers with units digit 0 0. If the units digit is 5 5, then there are four choices ( 1, 2, 3, 4 ) (1,2,3,4) for the thousands digit and there are 4 ⋅ 3 = 12 4⋅3=12 ways to complete the number. So there are 4 ⋅ 12 = 48 4⋅12=48 possible numbers with units digit 5 5. Together there are 36 + 48 = 84 36+48=84 possible numbers. Answer: 𝐷 D .
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