A 5-digit number

A 5-digit number divisible by 3 is to be formed using the numbers 0, 1, 2, 3, 4 and 5 without repetition . Find the total number of ways in which this can be done.

The answer is 216.

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1 solution

Anthony Holm
Sep 19, 2016

For a number to be divisible by 3, the sum of its digits must be divisible by 3. This restriction means that digits not divisible by 3 must come in pairs, 1 or 4 with 2 or 5. Because we exclude only one digit, all four such numbers must be included, meaning only 0 or 3 can be excluded. If 3 is included there are simply 5!=120 ways. When 0 is included there are only 4*4!=96 ways because 0 can't begin the number. Thus there are 120+96=216 ways to create a 5 digit number divisible by 3 with the using the digits 0-5 up to one each.

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