Question

If $\begin{vmatrix}^{9}C_{4}&^{9}C_{5}&^{10}C_{r}\\ ^{10}C_{6}&^{10}C_{7}&^{11}C_{r+2}\\ ^{11}C_{8}&^{11}C_{9}&^{12}C_{r+4}\end{vmatrix}=0$, then the value of $r$ is equal to _______.

Answer 1

Given $\begin{vmatrix}^{9}C_{4}&^{9}C_{5}&^{10}C_{r}\\ ^{10}C_{6}&^{10}C_{7}&^{11}C_{r+2}\\ ^{11}C_{8}&^{11}C_{9}&^{12}C_{r+4}\end{vmatrix}=0$
Apply $C_{2} \rightarrow C_{1}+C_{2}$
$\begin{vmatrix}^{9}C_{4}&^{10}C_{5}&^{10}C_{r}\\ ^{10}C_{6}&^{10}C_{7}&^{11}C_{r+2}\\ ^{11}C_{8}&^{12}C_{9}&^{12}C_{r+4}\end{vmatrix}=0$
Value of the determinant = 0
$\Rightarrow $ column 2 is same as column 3
$\Rightarrow r=5$