The method of moments estimator sets the sample mean equal to the population mean.

So, in a way you have two n's.

By the way, that data doesn't look like it would come from a Bin(250,p).

But what you should do is compute the sample mean of 1 through 8

which is (8)(9)/2=45 divided by 8 or 4.5 and set that equal to 250p and solve for p.

Thus

This data is hardly from a bin(250,p) distribution.

Maybe more of a Poisson.