= Solution
For adjacent <binomial coefficients>, the <ratio>
$$
\frac{\binom n{k+1}}{\binom nk}=\frac{n-k}{k+1}
$$
is greater than $1$ when $k<(n-1)/2$, equal to $1$ at $k=(n-1)/2$, and less than $1$ afterwards. For odd $n$, the row is therefore increasing up to its two equal central entries and decreasing after them. The <maximum> is attained exactly at
$$
\boxed{k^*=\frac{n-1}{2}\quad\text{or}\quad k^*=\frac{n+1}{2}.}
$$
They are distinct, including when $n=1$, so $k^*$ is \b[not unique]. This is the <unimodality of binomial coefficients>.
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