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Suppose there are 4 people and each person has an associated birth month. How many ways are there so that at least 2 people share the same birth month?

My first instinct is that it's $12\cdot12\cdot11\cdot10$ but that doesn't seem right.

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$$(\text{The number of ways at least 2 people share the same birth month}) =$$ $$(\text{The number of all possible arrangements of birth months})-$$ $$(\text{The number of ways no 2 people share the same birth month})$$ $$=(12\times12\times12\times 12)-(12\times 11\times 10\times 9)=20736-11880=8856$$

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