2011 AMC 10A Problem 13

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

The units digit must be

2

or

8

Case on whether the hundreds digit is

2

or

5

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.

Problem 13 in other years