NDA 1 2024 Maths, question 19

Topic: Matrices and determinants

Q19

For what value of $n$ is the determinant

$\left| \begin{matrix} C(9,4) & C(9,3) & C(10,n-2) \\ C(11,6) & C(11,5) & C(12,n) \\ C(m,7) & C(m,6) & C(m+1,n+1) \end{matrix} \right|=0$

for every $m>n$ ?

Solution.

Answer: (c) $6$

Pascal's identity gives C(9,4)+C(9,3)=C(10,4), C(11,6)+C(11,5)=C(12,6), and C(m,7)+C(m,6)=C(m+1,7).
With n=6 the third column is the sum of the first two, so the determinant is zero for every allowed m.