# How many ways to fill the $N \times N$ board by nonnegative integers, such that sum of the numbers of each row and each column is $R$?

How many ways to fill the $4 \times 4$ board by nonnegative integers, such that sum of the numbers of each row and each column is $3$?

I wrote a brute-force and got $2008$ which seems to be the right answer. I was wondering what is the combinatorial way of solving this one? How about the general problem of $N\times N$ or $N\times K$ with sum a $R$?

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math.binghamton.edu/dennis/Birkhoff –  deinst Apr 2 '12 at 15:19
@deinst: What do you mean? –  Quixotic Apr 2 '12 at 15:20
@Fool The problem you ask (in the NxN case) is counting the number of lattice points in a Birkhoff polytope. The number is a polynomial in the row/column sum. They computed this polynomial for N up to 10. –  deinst Apr 2 '12 at 15:24
For a simpler approach see math.stackexchange.com/questions/69000/… –  deinst Apr 2 '12 at 15:26