Let (1+x)n=C0+C1x+C2x2+…+CnxnStatement-1: S=C0+C0+C1+C0+C1+C2+⋯+C0+C1+⋯+Cn−1=n2n−1Statement-2: ∑j=1n ∑i

A
STATEMENT-1 is True, STATEMENT-2 is True; STATEMENT-2 is a correct explanation for STATEMENT-1
B
STATEMENT-1 is True, STATEMENT-2 is True; STATEMENT-2 is NOT a correct explanation for STATEMENT-1
C
STATEMENT-1 is True, STATEMENT-2 is False
D
STATEMENT-1 is False, STATEMENT-2 is True

We can write

$S = n C_{0} + (n - 1) C_{1} + (n - 2) C_{2} + \dots + 1 C_{n - 1} + 0 C_{n}$

Using $C_{r} = C_{n - r}$ we can rewrite (1) as

$S = 0 C_{0} + 1 C_{1} + 2 C_{2} + \dots + (n - 1) C_{n - 1} + n C_{n}$

Adding (1) and (2) we obtain

$2 S = n [C_{0} + C_{1} + C_{2} + \dots + C_{n}] = n (2^{n})$

$\Rightarrow S = n (2^{n - 1})$

In the expression $\sum_{j = 1}^{n} \sum_{i < j} (C_{i} + C_{j})$ each $C_{i} (0 \leq i \leq n)$ occurs exactly n times. Thus

$\sum_{j = 1}^{n} \sum_{i < j} (C_{i} + C_{j}) = n \sum_{k = 0}^{n} C_{k} = n (2^{n})$

Let $(1 + x)^{n} = C_{0} + C_{1} x + C_{2} x^{2} + \dots + C_{n} x^{n}$
Statement-1: $S = C_{0} + (C_{0} + C_{1}) + (C_{0} + C_{1} + C_{2}) + \dots + (C_{0} + C_{1} + \dots + C_{n - 1}) = n (2^{n - 1})$
Statement-2: $\sum_{j = 1}^{n} \sum_{i < j} (C_{i} + C_{j}) = (n + 1) 2^{n}$