How many even numbers are there with three digits such that if 5 is one of the digits, then 7 is the next digit?

An even number can be formed by using one of the digits 0, 2, 4, 6 or 8 at the units place.

5 can not be at ten's place, since in that case 7 has to be placed at the units place. In that case, the number formed is not an even number.

When 5 is at the hundred's place, then 7 will be at the ten's place. The units place can be filled in 5 ways by using one of the digits 0, 2, 4, 6 or 8.

Number of even numbers in this case = 1 × 1 × 5 = 5

When 5 is not at the hundred's place, so the hundred's place can be filled in 8 ways (0 and 5 cannot be used at hundred's place).Ten's place can be filled in 9 ways (5 cannot be used at ten's place).Units place can be filled in 5 ways (0, 2, 4, 6, or 8 can occupy the even place).

Number of even number in this case = 8 × 9 × 5 = 360

Hence, by fundamental principle of addition,

Total number of even number = 360 + 5 = 365.