Let `S_1=sum_(j=1)^(10)j(j-1)^(10)C_j ,""S_2=sum_(j=1)^(10)j""^(10)C_i "andS"_"3"=sum_(j=1)^(10)j^2""^("10")"C"_"j"dot` Statement-1: `S_3=""55xx2^9` Statement-2: `S_1=""90xx2^8a n d""S_2=""10xx2^8` . (1) Statement-1 is true, Statement-2 is true; Statement-2 is not the correct explanation for Statement-1 (2) Statement-1 is true, Statement-2 is false (3) Statement-1 is false, Statement-2 is true (4) Statement-1 is true, Statement-2 is true; Statement-2 is the correct explanation for Statement-1

Options

A. Statement 1 is false, statement 2 is true
B. Statement 1 is true, statement 2 is true, Statement 2 is a correct explanation of Statement 1
C. Statement 1 is true, statement 2 is true, Statement 2 is not a correct explanation of Statement 1
D. Statement 1 is true, statement 2 is false

Answer

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

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

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

We can rewrite \( j(j-1) \) as \( j^2 - j \):
\[ S_1 = \sum_{j=1}^{10} (j^2 - j) \binom{10}{j} \]

This can be separated into two sums:
\[ S_1 = \sum_{j=1}^{10} j^2 \binom{10}{j} - \sum_{j=1}^{10} j \binom{10}{j} \]

Using the identity \( \sum_{j=0}^{n} j \binom{n}{j} = n \cdot 2^{n-1} \):
\[ \sum_{j=1}^{10} j \binom{10}{j} = 10 \cdot 2^{9} = 10 \cdot 512 = 5120 \]

For \( \sum_{j=1}^{10} j^2 \binom{10}{j} \), we can use the identity:
\[ \sum_{j=0}^{n} j^2 \binom{n}{j} = n(n-1)2^{n-2} + n2^{n-1} \]
For \( n=10 \):
\[ \sum_{j=0}^{10} j^2 \binom{10}{j} = 10 \cdot 9 \cdot 2^{8} + 10 \cdot 2^{9} = 90 \cdot 256 + 10 \cdot 512 = 23040 + 5120 = 28160 \]

Thus,
\[ S_1 = 28160 - 5120 = 23040 \]

Now, we can express \( S_1 \) in terms of powers of 2:
\[ S_1 = 90 \cdot 2^8 \]

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

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

Using the identity we mentioned earlier:
\[ S_2 = 10 \cdot 2^{9} = 5120 \]

We can express \( S_2 \) as:
\[ S_2 = 10 \cdot 2^8 \]

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

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

From the identity we used earlier:
\[ S_3 = 28160 \]

Expressing \( S_3 \) in terms of powers of 2:
\[ S_3 = 55 \cdot 2^9 \]

### Conclusion

Now we can evaluate the statements:

- **Statement 1**: \( S_3 = 55 \cdot 2^9 \) is **true**.
- **Statement 2**: \( S_1 = 90 \cdot 2^8 \) and \( S_2 = 10 \cdot 2^8 \) is **false** (since \( S_2 = 10 \cdot 2^9 \)).

Thus, the correct option is:
(2) Statement-1 is true, Statement-2 is false.