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Prove that of any 100 different twelve digit numbers (first digit cannot be zero) there are two of them with the same first and fifth digit.

I'm new to this principle and need some assistance. I've been trying to understand how to approach this problem, but no dice. I guess I understand that there are two "pigeons" in the same "box", but still need help going through this.

1 Answers 1

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HINT: There are $9$ possible first digits, $1,2,\dots,9$, and $10$ possible fifth digits, $0,1,2,\dots,9$, so there are $9\cdot10$ possible combinations of first and fifth digits. If you have more than $9\cdot10$ twelve-digit numbers, ...

In other words, the pigeons are the $100$ numbers, and the boxes are the $9\cdot10$ possible combinations of first and fifth digit.

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    @blutuu: You’re very welcome.2012-11-30