Question

$\begin{vmatrix}1&1&1\\\ ^{n}C_{1}&^{n+1}C_{1}&^{n+2}C_{1}\\\ ^{n}C_{2}&^{n+1}C_{2}&^{n+2}C_{2}\end{vmatrix} =$

• A. 0
• B. 1
• C. -1
• D. none of these.

Answer 1

Answer: (B)

1

Answer 2

Given determinant $= \begin{vmatrix}1&1&1\\\ n&n+1&n+2\\\ \frac{n\left(n-1\right)}{2}&\frac{\left(n+1\right)n}{2}&\frac{\left(n+2\right)\left(n+1\right)}{2}\end{vmatrix}$ operate $C_{3} \to C_{3} -C_{2}$ and $C_{2} \to C_{2} -C_{1}$ $= \begin{vmatrix}1&0&0\\\ n&1&1\\\ \frac{n\left(n-1\right)}{2}&n&n+1\end{vmatrix}$ $= 1\left(n+1-n\right)=1$