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There are over 300 people in a movie theater. Prove that there has to be a month of the year when at least 26 of the movie goers have a birthday on the same day.

How do I prove this by contradiction?

number of pigeons = 300 , number of pigeonholes = 365

$number of pigeons/number of pigeonholes = 2$

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You're looking at months, not days. So if there's less than $26$ people per month, then the number of people is less than or equal to $25\cdot 12 = 300$, which contradicts the assumption that there are over $300$ people.

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  • $\begingroup$ I'm kind of confused about how the total number of people were calculated from $25*12$. I know that $25$ is used for the contradiction purpose, that $25$ people have their birthday on the same day of the month. $\endgroup$ – user3067059 Apr 12 '17 at 3:51
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    $\begingroup$ We assumed that there is less than 26 people in each month. If we let $x_i$ be the number of people in month $i$, we know $x_i\leq 25$ and the total amount of people is $\sum_{i=1}^{12} x_i \leq \sum_{i=1}^{12} 25 = 12\cdot 25 = 300 $ $\endgroup$ – Lazy Lee Apr 12 '17 at 3:53

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