IIT JEE 1983 - 'Adapted' - Subjective to Multi-Correct Q3

We note that the coefficient $C_k$ is binomial coefficient $\displaystyle {n \choose k}$ . Using the identity below:

$\begin{aligned} \left(\sum_{i=0}^n C_i \right)^2 & = \sum_{i=0}^n C_i^2 + 2 \sum_{i=0}^n \sum_{j=i+1}^n C_iC_j \\ \implies \sum _{i=0}^n \sum_{j=i+1}^n C_iC_j & = \frac 12 \left[ \left(\sum_{i=0}^n C_i \right)^2 - \sum_{i=0}^n C_i^2 \right] \\ & = \frac 12 \left[ \left( \sum_{i=0}^n {n \choose i} \right)^2 - {\color{#3D99F6} \sum_{i=0}^n {n \choose i}^2} \right] & \small \color{#3D99F6} \text{Vandermonde's identity: } \sum_{k=0}^r {m \choose k}{n \choose r-k} = {m+n \choose r} \\ & = \frac 12 \left[ 2^{2n} - \color{#3D99F6} {2n \choose n} \right] \\ & = 2^{2n-1} - \frac {(2n)!}{2(n!)^2} \end{aligned}$

Therefore,

• A: $\ a = 2n - 1$ true
• B: $\ b = 2n$ true
• C: $\ c = 2$ true
• D: $\ d = n$ true

And the answer is $\boxed{1111}$ .

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Reference: Vandermonde's identity