# 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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