Question

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

$12$

$20$

$72$

$120$

$200$

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.