 Solution

For adjacent binomial coefficients, the ratiois 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 atThey are distinct, including when $n=1$, so $k^{\ast }$ is not unique. This is the unimodality of binomial coefficients.