Statement-1: The number of divisors of 10! Is 280.
Statement-2: 10!=`2^(p)*3^(q)*5^(r)*7^(s)`, where p,q,r,s`in`N.

Options

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

Explanation

To solve the problem, we need to analyze both statements regarding \(10!\) (10 factorial) and determine their validity.

### Step-by-Step Solution:

**Step 1: Calculate \(10!\)**
First, we need to compute \(10!\):
\[ 10! = 10 \times 9 \times 8 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1 \]

**Step 2: Prime Factorization of \(10!\)**
Next, we will express \(10!\) in terms of its prime factors. We can break it down as follows:
- \(10 = 2^1 \times 5^1\)
- \(9 = 3^2\)
- \(8 = 2^3\)
- \(7 = 7^1\)
- \(6 = 2^1 \times 3^1\)
- \(5 = 5^1\)
- \(4 = 2^2\)
- \(3 = 3^1\)
- \(2 = 2^1\)

Now we can combine these:
\[ 10! = 2^1 \times 5^1 \times 3^2 \times 2^3 \times 7^1 \times 2^1 \times 3^1 \times 5^1 \times 2^2 \times 3^1 \times 2^1 \]

**Step 3: Count the Powers of Each Prime Factor**
Now we will sum the powers of each prime factor:
- For \(2\):
- From \(10\): \(1\)
- From \(8\): \(3\)
- From \(6\): \(1\)
- From \(4\): \(2\)
- From \(2\): \(1\)

Total for \(2\): \(1 + 3 + 1 + 2 + 1 = 8\)

- For \(3\):
- From \(9\): \(2\)
- From \(6\): \(1\)
- From \(3\): \(1\)

Total for \(3\): \(2 + 1 + 1 = 4\)

- For \(5\):
- From \(10\): \(1\)
- From \(5\): \(1\)

Total for \(5\): \(1 + 1 = 2\)

- For \(7\):
- From \(7\): \(1\)

Total for \(7\): \(1\)

Thus, we can express \(10!\) as:
\[ 10! = 2^8 \times 3^4 \times 5^2 \times 7^1 \]

**Step 4: Calculate the Number of Divisors**
The formula for the number of divisors \(d(n)\) of a number \(n = p_1^{e_1} \times p_2^{e_2} \times ... \times p_k^{e_k}\) is given by:
\[ d(n) = (e_1 + 1)(e_2 + 1)...(e_k + 1) \]

Applying this to our factorization:
\[ d(10!) = (8 + 1)(4 + 1)(2 + 1)(1 + 1) = 9 \times 5 \times 3 \times 2 \]

Calculating this step-by-step:
- \(9 \times 5 = 45\)
- \(45 \times 3 = 135\)
- \(135 \times 2 = 270\)

Thus, the number of divisors of \(10!\) is \(270\).

**Step 5: Validate Statements**
- **Statement 1**: The number of divisors of \(10!\) is \(280\). This is **false** because we calculated it to be \(270\).
- **Statement 2**: \(10! = 2^p \times 3^q \times 5^r \times 7^s\) where \(p, q, r, s \in \mathbb{N}\). This is **true** since we found \(p = 8\), \(q = 4\), \(r = 2\), \(s = 1\).

### Conclusion:
- Statement 1 is false.
- Statement 2 is true.

Thus, the correct answer is that Statement 1 is false and Statement 2 is true.