devarshi-dt-logo

Question:

Let S1 = ∑10j=1 j 10Cj, S2 = ∑10j=1 10Cj and S3 = ∑10j=1 j 2 10Cj. Statement-1: S3 = 55 × 29. Statement-2: S1 = 90 × 28 and S2 = 10 × 28. Choose the correct option: Statement-1 is true, Statement-2 is true; Statement-2 is not the correct explanation for Statement-1; Statement-1 is true, Statement-2 is false; Statement-1 is false, Statement-2 is true; Statement-1 is true, Statement-2 is true; Statement-2 is the correct explanation for Statement-1

Statement-1 is true, Statement-2 is true; Statement-2 is not the correct explanation for Statement-1

Statement-1 is false, Statement-2 is true

Statement-1 is true, Statement-2 is true; Statement-2 is the correct explanation for Statement-1

Statement-1 is true, Statement-2 is false

Solution:

S1 = ∑10j=1 j 10Cj = ∑10j=1 j ⁡⁡10!/(j!(10-j)!) = 90 × 28
S2 = ∑10j=1 10Cj = 210 -1 = 1023
S3 = ∑10j=1 j2 10Cj = ∑10j=1 j(10-j+j)10Cj = ∑10j=1j(10-j)10Cj + ∑10j=1 j2 10Cj
= ∑10j=110(j-1)10-1Cj-1+ ∑10j=1j2 10Cj
= 90 × 28 + 10 × 28 = 100 × 28 = 25600
S3 = 55 × 29 = 55 × 512 = 28160
Statement 1 is false, Statement 2 is true. Therefore, option B is correct.