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Let $n, l \in \mathbb{N}, l\leq n$ be fixed. Let $k\in \mathbb{N}$ with $0 \leq k \leq l$. How to show the following?

$${2n-l\choose n-k}\leq {2n-l \choose \frac{2n-l}{2}}$$

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    Is $l$ even?${}$2012-11-04
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    No, $l$ is any.2012-11-04
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    So you're using the Gamma function definition of $\binom{7}{7/2}$?2012-11-04
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    Yes, or just a Stirling formula.2012-11-04

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HINT

Prove that $$\displaystyle \begin{cases} \binom{2N}{r} < \binom{2N}{r+1} & \text{when } r \leq N-1\\ \binom{2N}{r+1} < \binom{2N}{r} & \text{when } r \geq N \end{cases}$$

We have that $$\binom{2N}{r+1} = \binom{2N}{r} \times \left( \frac{2N-r}{r+1} \right).$$ Note that $$ \begin{cases} \left( \frac{2N -r}{r+1} \right) > 1 & r \leq N - 1\\ \left( \frac{2N -r}{r+1} \right) < 1 & r \geq N \end{cases}$$ Hence, we have that $$\begin{cases} \binom{2N}{r} < \binom{2N}{r+1} & \text{when } r \leq N-1\\ \binom{2N}{r+1} < \binom{2N}{r} & \text{when } r \geq N \end{cases}$$

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    Tank you, but I stil don't see how to use it for my inequality. Could you elaborate, please. Thank you.2012-11-04
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    @user202312 In your case, choose $N = 2n-l$. Essentially, you want to prove that the middle binomial coefficient is the largest.2012-11-04
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Hint: Calculate

$$\frac{{2n-l\choose m+1}}{{2n-l\choose m}}$$

Can you find out when the fraction is more/less than 1?