SOLUTION: 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?

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Question 1119990: 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 by solver91311(24713) About Me  (Show Source):
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If a positive integer is divisible by 5, then the 1s digit must be either 5 or 0.

Consider the 4-digit positive integers with different digits, leading digit not zero, and 5 is the largest digit, where the 1s digit is zero.

There are 5 possibilities for the 1000s digit, the numbers 1 through 5, then for each of those possibilities there are four choices for the 100s digit, that is the numbers 1 through 5 less the digit that was selected for the 1000s digit. Then there are 3 ways to pick the 10s digit, 1 through 5 less the two that have already been used, and then the 1s digit is zero. So there are 5 times 4 times 3 different 4 digit numbers in this category, a total of 60 different numbers.

Now consider the 4-digit positive integers with different digits, leading digit not zero, and 5 is the largest digit, where the 1s digit is 5.

There are 4 ways to choose the 1000s digit, the numbers 1 through 4; zero not allowed as the lead digit, and 5 will be used as the 1s digit. Then there are 4 ways to choose the 100s digit, 0 through 4 less the one selected for the 1000s digit. Then there are 3 ways to choose the 10s digit and the 1s digit is 5. So 4 times 4 times 3 = 48.

Grand total: 108 different 4-digit numbers, lead digit not zero, 5 the largest digit, and divisible by 5.


John

My calculator said it, I believe it, that settles it