Of all the four-digit positive integers containing only digits from the set {2, 4, 6, 8}, what fraction of them have at least one of their digits repeated? Express your answer as a common fraction.

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- Feb 5th 2010, 09:17 AMsri340digits
Of all the four-digit positive integers containing only digits from the set {2, 4, 6, 8}, what fraction of them have at least one of their digits repeated? Express your answer as a common fraction.

- Feb 5th 2010, 09:46 AMArchie Meade
Hi sri340,

We can ask "how many have__no__repeated digits" ?

This is 4! as all 4 digits must be different.

There are $\displaystyle 4^4$ 4-digit numbers given the digits may be repeated.

Hence $\displaystyle 4^4-4!$ contain__at least one__repeated digit.

The fraction is $\displaystyle \frac{4^4-4!}{4^4}=\frac{4(4^3-6)}{4(4^3)}=\frac{4^3-6}{4^3}$