2011 AMC 10A Problems/Problem 13

Problem 13

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}?

$\text{(A)}\,12 \qquad\text{(B)}\,20 \qquad\text{(C)}\,72 \qquad\text{(D)}\,120 \qquad\text{(E)}\,200$

Solution

We split up into cases of the hundreds digits being $2$ or $5$ . If the hundred digits is $2$ , then the units digits must be $8$ in order for the number to be even and then there are $4$ remaining choices ( $1,5,7,9$ ) for the tens digit, giving $1 \times 1 \times 4=4$ possibilities. Similarly, there are $1 \times 2 \times 4=8$ possibilities for the $5$ case, giving a total of $\boxed{4+8=12 \ \mathbf{(A)}}$ possibilities.

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2011 AMC 10A Problems/Problem 13

Problem 13

Solution