STATEMENT -1: The tens digit of 1! + 2! + 3! + 4! + 5! +........... + 50! is 1.
STATEMENT-2 : The sum of divisors of `2^(4)3^(5)5^(2)7^(3)` is `2^(5).3^(6).5^(3).7^(4) - 2.3.5.7.`

Options

A. (a)Statement-1 is true ,statement -2 is true , statement -2 is a correct explanation for statement -3
B. (b)Statement -1 is true , statement -2 is true , statement -2 is NOT a correct explanation for statement -3
C. (c)Statement -1 is true , statement -2 is false
D. (d)Statement -1 is false , statement -2 is true

Answer

To solve the problem, we need to analyze both statements separately.

### Statement 1: The tens digit of \(1! + 2! + 3! + 4! + 5! + \ldots + 50!\) is 1.

1. **Calculate Factorials**:
- \(1! = 1\)
- \(2! = 2\)
- \(3! = 6\)
- \(4! = 24\)
- \(5! = 120\)
- \(6! = 720\)
- For \(n \geq 10\), \(n!\) will end with at least two zeros (because \(10! = 3628800\) and higher factorials will have more trailing zeros).

2. **Sum of Factorials**:
- We only need to consider the factorials from \(1!\) to \(9!\) since \(10!\) and above contribute \(0\) to the tens digit.
- Calculate the sum:
\[ 1 + 2 + 6 + 24 + 120 + 720 + 5040 + 40320 + 362880 \]
- Adding these values:
- \(1 + 2 = 3\)
- \(3 + 6 = 9\)
- \(9 + 24 = 33\)
- \(33 + 120 = 153\)
- \(153 + 720 = 873\)
- \(873 + 5040 = 5913\)
- \(5913 + 40320 = 46233\)
- \(46233 + 362880 = 409113\)

3. **Identify the Tens Digit**:
- The total sum is \(409113\).
- The tens digit is \(1\).

Thus, **Statement 1 is true**.

### Statement 2: The sum of divisors of \(2^4 \cdot 3^5 \cdot 5^2 \cdot 7^3\) is \(2^5 \cdot 3^6 \cdot 5^3 \cdot 7^4 - 2 \cdot 3 \cdot 5 \cdot 7\).

1. **Calculate the Sum of Divisors**:
- The formula for the sum of divisors \( \sigma(n) \) for \( n = p_1^{k_1} \cdot p_2^{k_2} \cdots p_m^{k_m} \) is:
\[ \sigma(n) = (1 + p_1 + p_1^2 + \ldots + p_1^{k_1})(1 + p_2 + \ldots + p_2^{k_2}) \cdots (1 + p_m + \ldots + p_m^{k_m}) \]
- For \(2^4\):
\[ 1 + 2 + 4 + 8 + 16 = 31 \]
- For \(3^5\):
\[ 1 + 3 + 9 + 27 + 81 + 243 = 364 \]
- For \(5^2\):
\[ 1 + 5 + 25 = 31 \]
- For \(7^3\):
\[ 1 + 7 + 49 + 343 = 400 \]

2. **Combine the Results**:
- Now, multiply these sums:
\[ 31 \cdot 364 \cdot 31 \cdot 400 \]

3. **Simplifying Statement 2**:
- The statement claims that this product equals \(2^5 \cdot 3^6 \cdot 5^3 \cdot 7^4 - 2 \cdot 3 \cdot 5 \cdot 7\).
- We need to check if the left-hand side equals the right-hand side.

4. **Conclusion**:
- After performing the calculations, we find that the left-hand side does not equal the right-hand side.
- Therefore, **Statement 2 is false**.

### Final Conclusion:
- **Statement 1 is true** and **Statement 2 is false**. Thus, the correct option is **C**.