AMC8 2015

AMC8 2015 · Q10

AMC8 2015 · Q10. It mainly tests Permutations.

How many integers between 1000 and 9999 have four distinct digits?

1000 到 9999 之间有多少个四位数具有四个不同数字？

(A) 3024 3024

(B) 4536 4536

(D) 6480 6480

(E) 6561 6561

Answer

Correct choice: (B)

正确答案：（B）

Solution

Answer (B): The thousands position can be filled by the digits $1$ through $9$ ($0$ is excluded). Without repetition, the hundreds position can be filled with any of the remaining $9$ digits (including $0$). Similarly without repetition, the tens position and ones position can be filled with the remaining $8$ and $7$ digits, respectively. Thus there are $9\cdot9\cdot8\cdot7=4536$ integers between $1000$ and $9999$ that have distinct digits.

答案（B）：千位可以用数字 $1$ 到 $9$ 填写（不包括 $0$）。在不重复的情况下，百位可以用剩下的 $9$ 个数字中的任意一个（包括 $0$）。同样在不重复的情况下，十位和个位分别可以用剩下的 $8$ 个和 $7$ 个数字来填写。因此，在 $1000$ 到 $9999$ 之间，数字各位互不相同的整数共有 $9\cdot9\cdot8\cdot7=4536$ 个。

Topics

CP.002 Permutations