Fun "facts" about integers

For non-negative integers \(a,b,c\) with their sum of 10, let the triplets \((a_1,b_1,c_1), (a_2,b_2, c_2) , \ldots, (a_n, b_n, c_n) \) denote the ordered triplets of \((a,b,c) \) such that \(\frac{10!}{a!b!c!}\) is an integer.

Find the value of \( \displaystyle \sum_{j=1}^n \frac{10!}{(a_j)!(b_j)!(c_j)!} \).

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