`S_(1)= sum_(j=1)^(10) j (j -1)""^(10)C_(j) and S_(2)= sum_(j=1)^(10)j.""^(10)C_(j) .` Statement-1 `S_(3) = 50xx2^(9)`.
Statement-2 `S_(1) = 90xx2^(8) and S_(2) = 10 xx 2^(8)`

Explanation

To solve the problem, we need to evaluate the sums \( S_1 \) and \( S_2 \) and check the validity of the statements provided.

### Step 1: Evaluate \( S_2 \)

We start with the expression for \( S_2 \):
\[ S_2 = \sum_{j=1}^{10} j \cdot \binom{10}{j} \]

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

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

Thus, we have:
\[ S_2 = 10 \cdot 2^9 \]

### Step 2: Evaluate \( S_1 \)

Now we evaluate \( S_1 \):
\[ S_1 = \sum_{j=1}^{10} j(j-1) \cdot \binom{10}{j} \]

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

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

Thus, we have:
\[ S_1 = 90 \cdot 2^8 \]

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

Now we evaluate \( S_3 \):
\[ S_3 = \sum_{j=1}^{10} j \cdot 2 \cdot \binom{10}{j} \]

This can be expressed as:
\[ S_3 = 2 \cdot \sum_{j=1}^{10} j \cdot \binom{10}{j} = 2 \cdot S_2 \]
Using the value of \( S_2 \):
\[ S_3 = 2 \cdot (10 \cdot 2^9) = 20 \cdot 2^9 \]

### Conclusion

Now we have:
- \( S_1 = 90 \cdot 2^8 \)
- \( S_2 = 10 \cdot 2^9 \)
- \( S_3 = 20 \cdot 2^9 \)

### Verification of Statements

**Statement 1:** \( S_3 = 50 \cdot 2^9 \) is **False** (it should be \( 20 \cdot 2^9 \)).

**Statement 2:** \( S_1 = 90 \cdot 2^8 \) and \( S_2 = 10 \cdot 2^8 \) is **False** (it should be \( S_2 = 10 \cdot 2^9 \)).