Question

The value of the determinant $\left|\begin{array}{ccc}1& {}^m{C}_1& {}^m{C}_2\\ 1& {}^{m+1}{C}_1& {}^{m+1}{C}_2\\ 1& {}^{2+m}{C}_1& {}^{m+2}{C}_2\end{array}\right|$ equals

• A. $0$
• B. $-1$
• C. $m!$
• D. $1$

Answer 1

The correct option is D $1$

$\left|\begin{array}{ccc}1& {}^m{C}_1& {}^m{C}_2\\ 1& {}^{m+1}{C}_1& {}^{m+1}{C}_2\\ 1& {}^{2+m}{C}_1& {}^{m+2}{C}_2\end{array}\right|$
$\left|\begin{array}{ccc}1& m& \frac{m(m-1)}{2}\\ 1& m+1& \frac{(m+1)m}{2}\\ 1& m+2& \frac{(m+2)(m+1)}{2}\end{array}\right|$

$R_3\to R_3-R_2,R_2\to R_2-R_1$

$=\left|\begin{array}{ccc}1& m& \frac{m(m-1)}{2}\\ 0& 1& m\\ 0& 1& m+1\end{array}\right|$

$=m+1-m=1$