The value of the expression ∑ 0 ≤ i < j ≤ n ( − 1 ) i + j − 1 n C i ⋅ n C j =

Let the required value be S i = 0 ∑ n j = 0 ∑ n ( − 1 ) i + j − 1 n C i n C j = i = 0 ∑ n ( − 1 ) 2 i − 1 ( n C i ) 2 + 2 S 0 = − i = 0 ∑ n ( n C i ) 2 + 2 S S = 2 n − 1 C n − 1 = 2 n − 1 C n

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Mathematics - Binomial Theorem Question with Solution | TestHub

MathematicsBinomial TheoremProperties of Binomial coefficientsMedium2 minQB

MathematicsMediumsingle choice

The value of the expression $\sum_{0 \leq i < j \leq n} (- 1)^{i + j - 1}^{n} C_{i} \cdot^{n} C_{j} =$

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Solution:

Let the required value be S $i = 0 \sum n j = 0 \sum n (- 1)^{i + j - 1}^{n} C_{i}^{n} C_{j} = i = 0 \sum n (- 1)^{2 i - 1} (^{n} C_{i})^{2} + 2 S 0 = - i = 0 \sum n (^{n} C_{i})^{2} + 2 S S =^{2 n - 1} C_{n - 1} =^{2 n - 1} C_{n}$

Stream:JEESubject:MathematicsTopic:Binomial TheoremSubtopic:Properties of Binomial coefficients

⏱ 2mℹ️ Source: QB

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