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. Statement-1 is true ,statement -2 is true , statement -2 is a correct explanation for statement -3
B. Statement -1 is true , statement -2 is true , statement -2 is NOT a correct explanation for statement -3
C. Statement -1 is true , statement -2 is false
D. Statement -1 is false , statement -2 is true

Answer

To solve the given statements, we will analyze each statement step by step.

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

1. **Calculate Factorials**:
- \(1! = 1\)
- \(2! = 2\)
- \(3! = 6\)
- \(4! = 24\)
- \(5! = 120\)
- \(6! = 720\)
- \(7! = 5040\)
- \(8! = 40320\)
- \(9! = 362880\)
- \(10! = 3628800\) (and all higher factorials will end with at least two zeros)

2. **Sum the Factorials**:
- From \(1!\) to \(9!\), we sum:
\[ 1 + 2 + 6 + 24 + 120 + 720 + 5040 + 40320 + 362880 = 403791 \]
- From \(10!\) to \(50!\), since they all end with at least two zeros, they do not contribute to the tens digit.

3. **Identify the Tens Digit**:
- The sum \(403791\) has a tens digit of \(9\).

4. **Conclusion for Statement 1**:
- The tens digit of \(1! + 2! + 3! + \ldots + 50!\) is \(9\), not \(1\). Therefore, Statement 1 is **False**.

### 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. **Sum of Divisors Formula**:
- The sum of divisors for \(n = p_1^{k_1} \cdot p_2^{k_2} \cdots p_m^{k_m}\) is given by:
\[ \sigma(n) = (1 + p_1 + p_1^2 + \ldots + p_1^{k_1})(1 + p_2 + \ldots + p_2^{k_2}) \cdots (1 + p_m + \ldots + p_m^{k_m}) \]

2. **Apply the Formula**:
- 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 \(7^3\):
\[ 1 + 7 + 49 + 343 = 400 \]

3. **Calculate the Total Sum of Divisors**:
- Now, multiply these results:
\[ \sigma(n) = 31 \cdot 364 \cdot 31 \cdot 400 \]

4. **Check the Given Expression**:
- The expression given in Statement 2 is:
\[ 2^5 \cdot 3^6 \cdot 5^3 \cdot 7^4 - 2 \cdot 3 \cdot 5 \cdot 7 \]
- We need to verify if this matches with our calculated sum of divisors.

5. **Conclusion for Statement 2**:
- After calculating and simplifying, we find that the expression does not hold true. Therefore, Statement 2 is also **False**.

### Final Conclusion:
- Both Statement 1 and Statement 2 are **False**.