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Question
Prove that the product of 2n consecutive negative integers is divisible by (2n)!
Solution
Let
\[ = \frac{\left( r + 2n - 1 \right)!}{\left( r - 1 \right)!}\]
\[ = \frac{\left( r + 2n - 1 \right)!}{\left( r - 1 \right)!\left( 2n \right)!} \times \left( 2n \right)!\]
\[ = {}^{r + 2n - 1} C_{2n} \times \left( 2n \right)!\]
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