Exercise 405Combinatorics › Binomial coefficientsHigh schoolAdded 3 Nov 2026

Monotonicity of binomial row

#monotonicity#binomial#ratio

Statement

Show that $\binom{n}{0}<\binom{n}{1}<\cdots<\binom{n}{\lfloor n/2\rfloor}$ and $\binom{n}{\lfloor n/2\rfloor}>\cdots>\binom{n}{n}$ .

Hint

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

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Ratio. $\binom{n}{k+1}/\binom{n}{k}=(n-k)/(k+1)>1$ iff $k<(n-1)/2$ .

Conclusion. The sequence increases up to the middle, then decreases.