Question

The value of $\left|\begin{array}{ccc}{}^{10}{C}_4& {}^{10}{C}_5& {}^{11}{C}_m\\ {}^{11}{C}_6& {}^{11}{C}_7& {}^{12}{C}_{m+2}\\ {}^{12}{C}_8& {}^{12}{C}_9& {}^{13}{C}_{m+4}\end{array}\right|=0$ when m is equal to:

• A. 6
• B. 5
• C. 4
• D. 1
• E. 2

Answer 1

Answer: (B)

$\left|\begin{array}{ccc}{}^{10}{C}_4& {}^{10}{C}_5& {}^{11}{C}_m\\ {}^{11}{C}_6& {}^{11}{C}_7& {}^{12}{C}_{m+2}\\ {}^{12}{C}_8& {}^{12}{C}_9& {}^{13}{C}_{m+4}\end{array}\right|=0$ Applying $R_2\to R_1+R_2$ $\Rightarrow$ $\left|\begin{array}{ccc}{}^{10}{C}_4{+}^{10}{C}_5& {}^{10}{C}_5& {}^{11}{C}_m\\ {}^{11}{C}_6{+}^{11}{C}_7& {}^{11}{C}_7& {}^{12}{C}_{m+2}\\ {}^{12}{C}_8{+}^{12}{C}_9& {}^{12}{C}_9& {}^{13}{C}_{m+4}\end{array}\right|=0$ $\Rightarrow$ $\left|\begin{array}{ccc}{}^{11}{C}_5& {}^{10}{C}_5& {}^{11}{C}_m\\ {}^{12}{C}_7& {}^{11}{C}_7& {}^{12}{C}_{m+2}\\ {}^{13}{C}_9& {}^{12}{C}_9& {}^{13}{C}_{m+4}\end{array}\right|=0$ It means either two rows or two columns are identical $\therefore$ ${}^{11}{C}_5{=}^{11}{C}_m{,}^{12}{C}_7{=}^{12}{C}_{m+2},$ ${}^{13}{C}_4{=}^{13}{C}_{m+4}$ $\Rightarrow$ $m=5$