Question

If $\left|\begin{array}{ccc}{}^9{C}_4& {}^9{C}_5& {}^{10}{C}_r\\ {}^{10}{C}_6& {}^{10}{C}_7& {}^{11}{C}_{r+2}\\ {}^{11}{C}_8& {}^{11}{C}_9& {}^{12}{C}_{r+4}\end{array}\right|=0$, then the value of $r$ is equal to _______.

Answer 1

Answer: 5

Given $\left|\begin{array}{ccc}{}^9{C}_4& {}^9{C}_5& {}^{10}{C}_r\\ {}^{10}{C}_6& {}^{10}{C}_7& {}^{11}{C}_{r+2}\\ {}^{11}{C}_8& {}^{11}{C}_9& {}^{12}{C}_{r+4}\end{array}\right|=0$
Apply $C_2\to C_1+C_2$
$\left|\begin{array}{ccc}{}^9{C}_4& {}^{10}{C}_5& {}^{10}{C}_r\\ {}^{10}{C}_6& {}^{10}{C}_7& {}^{11}{C}_{r+2}\\ {}^{11}{C}_8& {}^{12}{C}_9& {}^{12}{C}_{r+4}\end{array}\right|=0$
Value of the determinant = 0
$\Rightarrow$ column 2 is same as column 3
$\Rightarrow r=5$