Another dash of symmetry .....

Let \(S\) be the set of all symmetric \(3\)x\(3\) matrices which have only \(0\)'s and \(1\)'s as entries. (The number of entries that can be \(0\) can be any integer from \(0\) to \(9\) inclusive, as is the case for the number of entries that can be \(1\).)

Let \(N(k)\) be the number of elements of \(S\) that have \(k\) \(0\)'s as entries. Find \(\displaystyle\sum_{k=0}^9 (N(k))^{2}\).

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