Let `S_(1) = sum_(j=1)^(10) j(j-1).""^(10)C_(j), S_(2) = sum_(j=1)^(10)j.""^(10)C_(j)`, and `S_(3) = sum_(j=1)^(10) j^(2).""^(10)C_(j)`.
Statement 1 : `S_(3) = 55 xx 2^(9)`.
Statement 2 : `S_(1) = 90 xx 2^(8)` and `S_(2) = 10 xx 2^(8)`.

Options

A. (a) Statement 1 is false, statement 2 is true.
B. (b) Statement 1 is true, statement 2 is true, statement 2 is a correct explanation for statement 1.
C. (c) Statement 1 is true, statement 2 is true, statement 2 is not a correct explanation for statement 2.
D. (d) Statement 1 is true, statement 2 is false.

Explanation

To solve the problem, we need to evaluate the sums \( S_1 \), \( S_2 \), and \( S_3 \) based on the definitions provided and verify the statements given.

### Step 1: Calculate \( S_1 \)

Given:
\[ S_1 = \sum_{j=1}^{10} j(j-1) \binom{10}{j} \]

Since \( j(j-1) \) is zero for \( j=1 \), we can start the summation from \( j=2 \):
\[ S_1 = \sum_{j=2}^{10} j(j-1) \binom{10}{j} \]

Using the identity \( j(j-1) \binom{n}{j} = n(n-1) \binom{n-2}{j-2} \), we can rewrite \( S_1 \):
\[ S_1 = 10 \cdot 9 \sum_{j=2}^{10} \binom{8}{j-2} \]

Changing the index of summation by letting \( k = j - 2 \):
\[ S_1 = 90 \sum_{k=0}^{8} \binom{8}{k} = 90 \cdot 2^8 \]

Thus,
\[ S_1 = 90 \cdot 256 = 23040 \]

### Step 2: Calculate \( S_2 \)

Given:
\[ S_2 = \sum_{j=1}^{10} j \binom{10}{j} \]

Using the identity \( j \binom{n}{j} = n \binom{n-1}{j-1} \):
\[ S_2 = 10 \sum_{j=1}^{10} \binom{9}{j-1} \]

Changing the index of summation by letting \( k = j - 1 \):
\[ S_2 = 10 \sum_{k=0}^{9} \binom{9}{k} = 10 \cdot 2^9 \]

Thus,
\[ S_2 = 10 \cdot 512 = 5120 \]

### Step 3: Calculate \( S_3 \)

Given:
\[ S_3 = \sum_{j=1}^{10} j^2 \binom{10}{j} \]

Using the identity \( j^2 \binom{n}{j} = n(n-1) \binom{n-2}{j-2} + n \binom{n-1}{j-1} \):
\[ S_3 = 10 \cdot 9 \sum_{j=2}^{10} \binom{8}{j-2} + 10 \sum_{j=1}^{10} \binom{9}{j-1} \]

We already calculated the sums:
\[ \sum_{j=2}^{10} \binom{8}{j-2} = 2^8 \]
\[ \sum_{j=1}^{10} \binom{9}{j-1} = 2^9 \]

Thus,
\[ S_3 = 90 \cdot 256 + 10 \cdot 512 = 23040 + 5120 = 28160 \]

### Final Values:
- \( S_1 = 90 \cdot 2^8 \)
- \( S_2 = 10 \cdot 2^9 \)
- \( S_3 = 55 \cdot 2^9 \)

### Verification of Statements:
- Statement 1: \( S_3 = 55 \cdot 2^9 \) is **True**.
- Statement 2: \( S_1 = 90 \cdot 2^8 \) and \( S_2 = 10 \cdot 2^9 \) is **False**.