jmc

The arithmetic mean equals the sum of the elements divided by the number of elements in the row. In Pascal's Triangle where the first row is $n=0$, row $n$ has $n+1$ elements and the sum of the elements is $2^n$, which makes the arithmetic mean $\frac{2^n}{n+1}$. Now we look for the value of $n$ that satisfies $51.2=\frac{2^n}{n+1}$. If $51.2(n+1)=2^n$ and $n$ is a non-negative integer, we know that $2^n>51.2\Rightarrow n\ge 6$. If $n=6$, $2^n=64$, but $51.2(7)$ is much greater than $64$ (we can estimate $51.2(7)$ with $50\cdot 7=350$). With $n=7$ and $n=8$, we also find that $51.2(n+1)$ is greater than $2^n$. When we try $n=9$, we get $51.2(10)=2^9=512$, which is true. The value of $n$ is $\boxed{9}$.