Exercise 808Combinatorics › Binomial coefficientsL1Added 3 Nov 2026

The largest binomial coefficient

#maximum binomial#middle coefficient#parity

Statement

Show that the largest binomial coefficient $\binom{n}{k}$ is $\binom{n}{\lfloor n/2\rfloor}=\binom{n}{\lceil n/2\rceil}$ .

Hint

Compare consecutive coefficients via $\binom{n}{k+1}/\binom{n}{k}=(n-k)/(k+1)$ .

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Solution

Ratio. $\binom{n}{k+1}/\binom{n}{k}=(n-k)/(k+1)$ .

Increasing then decreasing. The ratio is $>1$ iff $n-k>k+1$ , i.e. $k<(n-1)/2$ . So the coefficients increase up to $k=\lfloor n/2\rfloor$ and decrease after.

Conclusion. The maximum is $\binom{n}{\lfloor n/2\rfloor}$ .

Example. $n=6$ : max is $\binom{6}{3}=20$ .

The largest binomial coefficient

Statement

Solution

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