Question

在 $200$ 到 $700$ 之间，有多少个偶整数的各位数字都不同，且都来自集合 $\{1,2,5,7,8,9\}?$

How many even integers are there between $200$ and $700$ whose digits are all different and come from the set $\{1,2,5,7,8,9\}?$

$12$

$20$

$72$

$120$

$200$

百位只能是 $2$ 或 $5$ ，

情况一：百位是 $2$ 。

此时个位只能是 $8$ ，十位还有 $4$ 种选择。

这一情况得到 $1 \cdot 4 = 4$ 个数。

情况二：百位是 $5$ 。

同理，个位只能选 $2$ 或 $8$ ，十位还有 $4$ 种选择。

这一情况给出 $2 \cdot 4 = 8$ 个数。

因此整数总数为 $4 + 8 = 12$ 。

所以正确答案是 A。

Since the hundreds digit can only be a $2$ or $5,$ we can case on this value.

Case 1: hundreds digit is $2$

The only option for the units digit is $8,$ since the number must be even. This leaves $4$ options for the tens digit.

This gives us $1 \cdot 4 = 4$ numbers for this case.

Case 2: hundreds digit is $5$

Similarly to above, $2$ and $8$ are the only options for the units digit, leaving $4$ options for the tens digit.

This gives us $2 \cdot 4 = 8$ numbers for this case.

The total number of integers is then $4 + 8 = 12.$

Thus, A is the correct answer.