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Define S1 = ∑(j=1 to 10) j(j-1)   10Cj, S2 = ∑(j=1 to 10) j   10Cj, and S3 = ∑(j=1 to 10) j²   10Cj. Which of the following statements is correct? Statement 1: S3 = 55 × 2⁹. Statement 2: S1 = 90 × 2⁸ and S2 = 10 × 2⁸.

1. Statement 1 is correct, and Statement 2 is correct; however, Statement 2 does not correctly explain Statement 1.
2. Statement 1 is correct, and Statement 2 is incorrect.
3. Statement 1 is incorrect, and Statement 2 is correct.
4. Statement 1 is correct, and Statement 2 is correct; moreover, Statement 2 correctly explains Statement 1.

Correct answer: Statement 1 is correct, and Statement 2 is incorrect.

Solution

Statement 1 is accurate because it correctly calculates S3 using combinatorial identities, while Statement 2 incorrectly computes S1 and S2, leading to the conclusion that Statement 1 stands alone as correct.

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