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7.Binomial Theorem
normal
$\sum \limits_{\substack{i, j=0 \\ i \neq j}}^{ n }{ }^n C_i{ }^n C_j$ बराबर है :
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
