If `n = 12, r = 4`, then evaluate the following:
(i) `(n!)/(r!(n-r)!)` (ii) `(n!)/((n-r+2)!)`

Answer

To solve the given problem step by step, we will evaluate both parts of the question:

Given:
- \( n = 12 \)
- \( r = 4 \)

### Part (i): Evaluate \( \frac{n!}{r!(n-r)!} \)

1. **Substitute the values of \( n \) and \( r \)**:
\[ \frac{n!}{r!(n-r)!} = \frac{12!}{4!(12-4)!} = \frac{12!}{4! \cdot 8!} \]

2. **Express \( 12! \) in terms of \( 8! \)**:
\[ 12! = 12 \times 11 \times 10 \times 9 \times 8! \]

3. **Substitute \( 12! \) back into the equation**:
\[ \frac{12 \times 11 \times 10 \times 9 \times 8!}{4! \cdot 8!} \]

4. **Cancel \( 8! \) from the numerator and denominator**:
\[ = \frac{12 \times 11 \times 10 \times 9}{4!} \]

5. **Calculate \( 4! \)**:
\[ 4! = 4 \times 3 \times 2 \times 1 = 24 \]

6. **Now substitute \( 4! \) back into the equation**:
\[ = \frac{12 \times 11 \times 10 \times 9}{24} \]

7. **Calculate the numerator**:
\[ 12 \times 11 = 132 \]
\[ 132 \times 10 = 1320 \]
\[ 1320 \times 9 = 11880 \]

8. **Now divide by 24**:
\[ = \frac{11880}{24} = 495 \]

**Final Answer for Part (i)**: \( 495 \)

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### Part (ii): Evaluate \( \frac{n!}{(n-r+2)!} \)

1. **Substitute the values of \( n \) and \( r \)**:
\[ \frac{n!}{(n-r+2)!} = \frac{12!}{(12-4+2)!} = \frac{12!}{10!} \]

2. **Express \( 12! \) in terms of \( 10! \)**:
\[ 12! = 12 \times 11 \times 10! \]

3. **Substitute \( 12! \) back into the equation**:
\[ = \frac{12 \times 11 \times 10!}{10!} \]

4. **Cancel \( 10! \) from the numerator and denominator**:
\[ = 12 \times 11 \]

5. **Calculate the product**:
\[ 12 \times 11 = 132 \]

**Final Answer for Part (ii)**: \( 132 \)

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### Summary of Answers:
- (i) \( 495 \)
- (ii) \( 132 \)