AMC8 2012

AMC8 2012 · Q10

AMC8 2012 · Q10. It mainly tests Basic counting (rules of product/sum), Casework.

How many 4-digit numbers greater than 1000 are there that use the four digits of 2012?

使用数字2012的四个数字，有多少个大于1000的四位数？

(A) 6 6

(B) 7 7

(D) 9 9

(E) 12 12

Answer

Correct choice: (D)

正确答案：（D）

Solution

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 $\frac{3!}{2!1!} \implies 3$ such numbers. When the leading digit is $2$, you can make $3! \implies 6$ such numbers. Summing the amounts of numbers, we find that there are $\boxed{\textbf{(D)}\ 9}$ such numbers.

对于这个问题，我们只需找出能由数字$2012$构成的有效四位数数量，因为所有有效四位数都大于$1000$。最好的方法是分类讨论。首位只能是$1$或$2$。当首位是$1$时，可以构成$\frac{3!}{2!1!} \implies 3$个这样的数。当首位是$2$时，可以构成$3! \implies 6$个这样的数。将数量相加，我们得到有$\boxed{\textbf{(D)}\ 9}$个这样的数。

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