JEE Mains · Maths · STD 11 - 7. binomial theoram
\(\sum_{\substack{i, j=0 \\ i \neq j}}^{n}{ }^{n} C_{i}{ }^{n} C_{j}\) 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 \(2^{ n -1}+{ }^{2 n -1} C _{ n }\)
Answer & Solution
Correct Answer
(A) \(2^{2 n }-{ }^{2 n } C _{ n }\)
Step-by-step Solution
Detailed explanation
\(\sum_{\substack{i, j=0 \\ i \neq j}}^{n}{ }^{n} C_{i}{ }^{n} C_{j}\) \(=\sum_{ i =0}^{ n }{ }^{ n } C _{ i } \cdot \sum_{ j =0}^{ n }{ }^{ n } C _{ j }-\sum_{ i = j =0}^{ n }\left({ }^{ n } C _{ i }\right)^{2}\) \(=\left(2^{ n }\right)\left(2^{ n }\right)-{ }^{2 n } C _{ n }\)…
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