Question

The value of $\Delta =\begin{vmatrix} ^5C_0 & ^5C_3 & 14 \\\[0.3em] ^5C_1 & ^5C_4 &1 \\\[0.3em] ^5C_2 & ^5C_5 & 1 \end{vmatrix}$ is

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

Answer 1

Answer: (B)

-576

Answer 2

$\Delta = \begin{vmatrix} ^5C_0 & ^5C_3 & 14 \\\[0.3em] ^5C_1 & ^5C_4 &1 \\\[0.3em] ^5C_2 & ^5C_5 & 1 \end{vmatrix}$ Operate $R_1 + R_2 + R_3$ = $\begin{vmatrix} ^5C_0 +^5C_1 +^5C_2& ^5C_3+^5C_4+^5C_5 & 14+1+1 \\\[0.3em] 5 & 5 &1 \\\[0.3em] 10 &1 & 1 \end{vmatrix}$ = $\begin{vmatrix}16&16&16\\\ 5&5&1\\\ 10&1&1\end{vmatrix}=16\begin{vmatrix}1&1&1\\\ 5&5&1\\\ 10&1&1\end{vmatrix}$ [$\because\,^5C_0+^5C_1+^5C_2 +^5C_3+^5C_4+^5C_5 = 16$ ] = $\begin{vmatrix}1&1&1\\\ 5&5&1\\\ 9&0&0\end{vmatrix}= 16 \times 9 \left(-4\right)=-576$