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  4. The value of Δ =| 5C0 5C3 14 [0.3em] 5C1 5C4 &1 [0.3em] 5C2 5C5 1 | is

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Q. 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

Determinants

Solution:

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