$The value of the determinant |\begin{array}{ccc} 1 & 1 & 1 \\ ^{m} C_{1} & ^{m + 1} C_{1} & ^{m + 2} C_{1} \\ ^{m} C_{2} & ^{m + 1} C_{2} & ^{m + 2} C_{2} \end{array}| is equal to$

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Detailed Solution

$\begin{array}{l} |\begin{array}{ccc} 1 & 1 & 1 \\ ^{m} C_{1} & ^{m + 1} C_{1} & ^{m + 2} C_{1} \\ ^{m} C_{2} & ^{m + 1} C_{2} & ^{m + 2} C_{2} \end{array}| \\ = |\begin{array}{ccc} 1 & 1 & 1 \\ ^{m} C_{1} & ^{m + 1} C_{1} & ^{m + 1} C_{0} +^{m + 1} C_{1} \\ ^{m} C_{2} & ^{m + 1} C_{2} & ^{m + 1} C_{1} +^{m + 1} C_{2} \end{array}| \\ = |\begin{array}{ccc} 1 & 1 & 0 \\ ^{m} C_{1} & ^{m + 1} C_{1} & ^{m + 1} C_{0} \\ ^{m} C_{2} & ^{m + 1} C_{2} & ^{m + 1} C_{1} \end{array}| {Applying C_{3} \to C_{3} - C_{2}\} \end{array}$
$\begin{array}{l} = |\begin{array}{ccc} 1 & 1 & 0 \\ ^{m} C_{1} & ^{m} C_{0} +^{m} C_{1} & ^{m + 1} C_{0} \\ ^{m} C_{2} & ^{m} C_{1} +^{m} C_{2} & ^{m + 1} C_{1} \end{array}| \\ = |\begin{array}{ccc} 1 & 0 & 0 \\ ^{m} C_{1} & ^{m} C_{0} & ^{m + 1} C_{0} \\ ^{m} C_{2} & ^{m} C_{1} & ^{m + 1} C_{1} \end{array}| Applying C_{2} \to C_{2} - C_{1}\} \\ =^{m} {C_{0}}^{m + 1} C_{1} -^{m + 1} {C_{0}}^{m} C_{1} \\ = m + 1 - m \\ = 1 \end{array}$