The value of ( l 30 0 ) , ( l 30 10 ) - ( l 30 1 ) , ( l 30 11 ) … - Vidyadip

MCQ

The value of $\left( \begin{array}{l}30\\0\end{array} \right)\,\left( \begin{array}{l}30\\10\end{array} \right) - \left( \begin{array}{l}30\\1\end{array} \right)\,\left( \begin{array}{l}30\\11\end{array} \right)$ + $\left( \begin{array}{l}30\\2\end{array} \right)\,\left( \begin{array}{l}30\\12\end{array} \right) + ....... + \left( \begin{array}{l}30\\20\end{array} \right)\,\left( \begin{array}{l}30\\30\end{array} \right)$

A
$^{60}{C_{20}}$
✓
$^{30}{C_{10}}$
C
$^{60}{C_{30}}$
D
$^{40}{C_{30}}$

✓

Answer

Correct option: B.

$^{30}{C_{10}}$

${(1 - x)^{30}} = {\,^{30}}{C_0}{x^0} - {\,^{30}}{C_1}{x^1} + {\,^{30}}{C_2}{x^2} + ...... + {( - 1)^{30}}{\;^{30}}{C_{30}}{x^{30}}.......(i)$
${(x + 1)^{30}} = {\,^{30}}{C_0}{x^{30}} + {\,^{30}}{C_1}{x^{29}} + {\,^{30}}{C_2}{x^{28}}+ ...... + {\,^{30}}{C_{10}}{x^{20}} + .... + {\,^{30}}{C_{30}}{x^0}........(ii)$
Multiplying $(i)$ and $(ii)$ and equating the coefficient of $x^{20}$ on both sides, we get required sum
$=$ coefficient of $x^{20}$ in $(1 -x^2)^{30}={^{30}}{C^{10}}$.

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