Σ_{substack{i, j=0 i ≠ j}}^{n}{ }^{n} C_{i}{ }^{n} C_{j} is equal to

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7.Binomial Theorem

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$\sum_{\substack{i, j=0 \\ i \neq j}}^{n}{ }^{n} C_{i}{ }^{n} C_{j}$ is equal to

A

$2^{2 n }-{ }^{2 n } C _{ n }$

B

$2^{2 n -1}-^{2 n -1} C _{ n -1}$

C

$2^{2 n }-\frac{1}{2}{ }^{2 n } C _{ n }$

D

$2^{ n -1}+{ }^{2 n -1} C _{ n }$

(JEE MAIN-2022)

Solution

$\sum_{\substack{i, j=0 \\ i \neq j}}^{n}{ }^{n} C_{i}{ }^{n} C_{j}$

$=\sum_{ i =0}^{ n }{ }^{ n } C _{ i } \cdot \sum_{ j =0}^{ n }{ }^{ n } C _{ j }-\sum_{ i = j =0}^{ n }\left({ }^{ n } C _{ i }\right)^{2}$

$=\left(2^{ n }\right)\left(2^{ n }\right)-{ }^{2 n } C _{ n }$

$=2^{2 n }-{ }^{2 n } C _{ n }$

Standard 11

Mathematics

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