Question

[amc8.2011.23] how many 4-digit positive integers have four different digits, where the leading digit is not zero, the integer is a multiple of 5, and 5 is the largest digit?

Answer 1

108

Explanation:

we have to consider two cases, one where the last digit is 5 and the other where the last digit is 0.

case 1: the last digit is 0

_ _ _ 0 and the other three could be anything from 1-5. this is the same as 5×4×3=60

case 2: the last digit is 5

_ _ _ 5. the first digit cannot be zero, so we have 4 possibilities. the next two have 4 and 3 possibilities respectively, totalling to 4×4×3=48

so, the total is 60+48=108