Question

Assertion (A): $S_3 = 55 \times 2^9$
Reason (R): $S_1 = 90 \times 2^8$ and $S_2 = 10 \times 2^8$

• A. Both (A) and (R) are true and R is the correct explanation of A
• B. Both (A) and (R) are true, but R is not the correct explanation of A
• C. (A) is true, but (R) is false
• D. (A) is false, but (R) is true

Hint:

Verify both assertion and reason separately with binomial sums; don't assume correctness of both.

Answer 1

The Correct Option is C

Calculate $S_1 = \sum_{j=1}^{10} j(j-1) . {10 \choose j}$, $S_2 = \sum_{j=1}^{10} j . {10 \choose j}$ and $S_3 = \sum_{j=1}^{10} j^2 . {10 \choose j}$.
Use binomial identities and algebraic simplifications:
$S_1 = 90 \times 2^8$ is false (the actual value differs), $S_2 = 10 \times 2^8$ is true, and $S_3 = 55 \times 2^9$ is true.
Since $S_1$ in Reason (R) is false, (R) is false even though (A) is true.