AMC 8 2012 Problem 10
AMC8 2012 年第 10 题
排列与组合★★☆☆☆
How many 4-digit numbers greater than 1000 are there that use the four digits of 2012?
使用 2012 的四个数字，能组成多少个大于 1000 的四位数？
- A.
- B.
- C.
- D.
- E.
正确答案 · Correct Answer
D
解析 · Solution
Solution 1
For this problem, all we need to do is find the amount of valid 4-digit numbers that can be made from the digits of 2012, since all of the valid 4-digit number will always be greater than 1000. The best way to solve this problem is by using casework. There can be only two leading digits, namely 1 or 2. When the leading digit is 1, you can make such numbers. When the leading digit is 2, you can make such numbers. Summing the amounts of numbers, we find that there are 9 such numbers.
Notice that the first digit cannot be 0, as the number is greater than 1000. Therefore, there are three digits that can be in the thousands. The rest three digits of the number have no restrictions, and therefore there are for each leading digit. Since the two 2 's are indistinguishable, there are such numbers .
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