Question

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

84 integers

Explanation:

Since the number is a multiple of 5, the units digit must be 0 or 5.

If 5 is the units digit:

The thousands digit can be 1,2,3, or 4 so there are 4 options. The hundreds digit also has 4 options because it can have 0. The tens digit has 3 options.

4*4*3=48

If 0 is the units digit:

Since 5 is the largest digit, there must be a 5 in the integer. If the 5 is the hundreds digit, there are 12 ways. There are 3 places the 5 could be so 12*3=36.

48+36=84 Integers

Answer 2

`5\cdot4\cdot3\cdot1+4\cdot4\cdot3\cdot1=60+48=108`