Question

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

• A. 6
• B. 4
• C. 5
• D. none of these.

Answer 1

Answer: (C)

$\Delta =\left|\begin{array}{ccc}{}^{10}{C}_4& {}^{10}{C}_5& {}^{11}{C}_m\\ {}^{11}{C}_6& {}^{11}{C}_7& {}^{12}{C}_{m+2}\\ {}^{12}{C}_8& {}^{12}{C}_9& {}^{13}{C}_{m+4}\end{array}\right|=0$
Operate $C_1+C_2$
$\Delta =\left|\begin{array}{ccc}{}^{10}{C}_4& {}^{10}{C}_4{+}^{10}{C}_5& {}^{11}{C}_m\\ {}^{11}{C}_6& {}^{11}{C}_6{+}^{11}{C}_7& {}^{12}{C}_{m+2}\\ {}^{12}{C}_8& {}^{12}{C}_8{+}^{12}{C}_9& {}^{13}{C}_{m+4}\end{array}\right|=0$
$=\left|\begin{array}{ccc}{}^{10}{C}_4& {}^{12}{C}_4& {}^{11}{C}_m\\ {}^{11}{C}_6& {}^{12}{C}_7& {}^{12}{C}_{m+2}\\ {}^{12}{C}_8& {}^{13}{C}_9& {}^{13}{C}_{m+4}\end{array}\right|=0$
Clearly, $m=5$ satisfies the above result, [$\because C_1,C_3$ will be identical]