Binomial Theorem And Its Simple Applications Question 182
Question: The value of $ C \begin{matrix} 30 \\ 0 \\ \end{matrix} C \begin{matrix} 30 \\ 10 \\ \end{matrix} -C \begin{matrix} 30 \\ 1 \\ \end{matrix} C \begin{matrix} 30 \\ 11 \\ \end{matrix} +C \begin{matrix} 30 \\ 2 \\ \end{matrix} C \begin{matrix} 30 \\ 12 \\ \end{matrix} +……+C \begin{matrix} 30 \\ 20 \\ \end{matrix} C \begin{matrix} 30 \\ 30 \\ \end{matrix} $
[IIT Screening 2005]
Options:
A) $ ^{60}C_{20} $
B) $ ^{30}C_{10} $
C) $ ^{60}C_{30} $
D) $ ^{40}C_{30} $
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Answer:
Correct Answer: B
Solution:
- $ {{(1-x)}^{30}}={{,}^{30}}C_0x^{0}-{{,}^{30}}C_1x^{1}+{{,}^{30}}C_2x^{2} $
$ +……+{{(-1)}^{30}}{{\ }^{30}}C_{30}x^{30} $ ….(i) $ {{(x+1)}^{30}}={{,}^{30}}C_0x^{30}+{{,}^{30}}C_1x^{29}+{{,}^{30}}C_2x^{28} $
$ +……+{{,}^{30}}C_{10}x^{20}+….+{{,}^{30}}C_{30}x^{0} $ ….(ii)
Multiplying (i) and (ii) and equating the coefficient of x20 on both sides,
we get required sum = coefficient of $x^{20}$ in $(1 - x^2)^{30}=^{30}C_{10}$.
BETA
