Counting four-digit numbers with repeating digits

Question

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 repeating?

Express your answer as a fraction.

I'm having trouble with this exercise.

Can someone help me get started?

Answer 1

All the $4$-digit numbers with digits among $\{2,4,6,8\}$ are $4^4=256$. All those which do not repeat digits are $4!=24$. So those with at least one repetition are $4^4-4!=232$. Their fraction is hence $$ \frac{232}{256}. $$

Answer 2

Usual tricks to get started for any problem

• Try a simpler problem
• Try brute-force calculation
• Look for other problems that this one resembles

Have you any ideas on how to do any of these three things?

For counting problems in particular, one trick that is:

• Try counting the complement: how many things are there in all, and how many aren't the kind of thing you're looking for?

If you are having any troubles at all, it's nearly always worth at least writing down the complementary problem to see if it offers any inspiration.