Question #41001

Let $S$ be a dataset consisting of $n+1$ observations $x_r = \binom{n}{r}$ for $r = 0, 1, 2, \dots, n$. It is given that the maximum value in $S$ is not unique, and in the binomial expansion of $(1+x)^n$ at $x = \frac{3}{4}$, the numerically greatest term is the $4^{\text{th}}$ term. A student calculates the mean and variance of the dataset $S$. While doing so, they misread exactly one observation of value $n$ as another integer $M$. If the incorrectly calculated mean is $17$, what is the incorrectly calculated variance?