For what value of n is the determinant \begin{vmatrix} C(9,4) & C(9,3) & C(10,n-2) \\ C(11,6) & C(11,5) & C(12,n) \\ C(m,7) & C(m,6) & C(m+1,n+1) \end{vmatrix} = 0 for every m \gt n?

Explanation

Using the property C(n, r) + C(n, r-1) = C(n+1, r), adding Column 1 and Column 2 gives values matching the pattern of Column 3. For the determinant to be identically zero, Column 3 must be exactly equal to C_1 + C_2. So C(9,4) + C(9,3) = C(10,4). Setting this equal to C(10, n-2) implies n-2 = 4, which means n = 6.

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