Question

The value of $\Delta =\left|\begin{array}{ccc}{}^5{C}_0& {}^5{C}_3& 14\\ {}^5{C}_1& {}^5{C}_4& 1\\ {}^5{C}_2& {}^5{C}_5& 1\end{array}\right|$ is

• A. 0
• B. -576
• C. 80
• D. none of these

Answer 1

Answer: (B)

$\Delta =\left|\begin{array}{ccc}{}^5{C}_0& {}^5{C}_3& 14\\ {}^5{C}_1& {}^5{C}_4& 1\\ {}^5{C}_2& {}^5{C}_5& 1\end{array}\right|$ Operate $R_1+R_2+R_3$
= $\left|\begin{array}{ccc}{}^5{C}_0{+}^5{C}_1{+}^5{C}_2& {}^5{C}_3{+}^5{C}_4{+}^5{C}_5& 14+1+1\\ 5& 5& 1\\ 10& 1& 1\end{array}\right|$
= $\left|\begin{array}{ccc}16& 16& 16\\ 5& 5& 1\\ 10& 1& 1\end{array}\right|=16\left|\begin{array}{ccc}1& 1& 1\\ 5& 5& 1\\ 10& 1& 1\end{array}\right|$ [$\because {}^5{C}_0{+}^5{C}_1{+}^5{C}_2{+}^5{C}_3{+}^5{C}_4{+}^5{C}_5=16$ ]
= $\left|\begin{array}{ccc}1& 1& 1\\ 5& 5& 1\\ 9& 0& 0\end{array}\right|=16\times 9\left(-4\right)=-576$