The value of binom{30}{0}binom{30}{10} - bino | vedclass

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Question

(B) We know that the given expression is the coefficient of $x^{20}$ in the product of two binomial expansions.Consider $(1-x)^{30} = \sum_{r=0}^{30}

Vedclass · Accepted Answer

$^{30}C_{10}$

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(B) We know that the given expression is the coefficient of $x^{20}$ in the product of two binomial expansions.Consider $(1-x)^{30} = \sum_{r=0}^{30} (-1)^r \binom{30}{r} x^r = \binom{30}{0} - \binom{30}{1}x + \binom{30}{2}x^2 - ....... + \binom{30}{20}x^{20} - ....... + \binom{30}{30}x^{30}$.Consider $(x+1)^{30} = \sum_{k=0}^{30} \binom{30}{k} x^{30-k} = \binom{30}{0}x^{30} + \binom{30}{1}x^{29} + ....... + \binom{30}{10}x^{20} + ....... + \binom{30}{30}$.The given expression is the coefficient of $x^{20}$ in the product $(1-x)^{30}(x+1)^{30} = (1-x^2)^{30}$.In the expansion of $(1-x^2)^{30} = \sum_{k=0}^{30} \binom{30}{k} (-x^2)^k = \sum_{k=0}^{30} \binom{30}{k} (-1)^k x^{2k}$.To find the coefficient of $x^{20}$,we set $2k = 20$,which gives $k = 10$.The coefficient is $\binom{30}{10} (-1)^{10} = \binom{30}{10}$.

The value of $\binom{30}{0}\binom{30}{10} - \binom{30}{1}\binom{30}{11} + \binom{30}{2}\binom{30}{12} - ....... + \binom{30}{20}\binom{30}{30}$ is:

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