Explanation:

- Problem Breakdown:

- The expression is a summation involving factorials:

- \(2(2 \times 1)\)

- \(3(3 \times 2 \times 1)\)

- \(4(4 \times 3 \times 2 \times 1)\)

- \(5(5 \times 4 \times 3 \times 2 \times 1)\)

- ...

- \(9(9 \times 8 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1)\)

- The terms simplify as:

- \(2 \times 2! + 3 \times 3! + 4 \times 4! + \ldots + 9 \times 9!\)

- This pattern is generalized to:

- \((n \times n!) = (n+1)! - n!\)

- Hence, the summation becomes:

- \((3! - 2!) + (4! - 3!) + \ldots + (10! - 9!)\)

- Which simplifies to \(10! - 2!\)

- Adding '2' gives \(10!-2!+2=10!\)

- Options:

- Option 1: 11!

- Option 2: 10! – This is the correct answer.

- Option 3: 10 + 10!

- Option 4: 11 + 10!