Evaluate `(n!)/(r!(n-r)!)` when :
(i) `n=6, r=2`
(ii) `n=7, r=4`
(iii) `n=15, r=12`.

Answer

To evaluate the expression \(\frac{n!}{r!(n-r)!}\) for the given values of \(n\) and \(r\), we will follow these steps for each case:

### (i) When \(n = 6\) and \(r = 2\)

1. **Substitute the values into the formula**:
\[ \frac{6!}{2!(6-2)!} = \frac{6!}{2! \cdot 4!} \]

2. **Calculate the factorials**:
- \(6! = 6 \times 5 \times 4!\)
- \(2! = 2 \times 1 = 2\)
- \(4! = 4!\) (remains as is)

3. **Substitute back into the equation**:
\[ \frac{6 \times 5 \times 4!}{2 \times 1 \times 4!} \]

4. **Cancel out \(4!\)**:
\[ \frac{6 \times 5}{2} = \frac{30}{2} = 15 \]

### (ii) When \(n = 7\) and \(r = 4\)

1. **Substitute the values into the formula**:
\[ \frac{7!}{4!(7-4)!} = \frac{7!}{4! \cdot 3!} \]

2. **Calculate the factorials**:
- \(7! = 7 \times 6 \times 5 \times 4!\)
- \(4! = 4!\) (remains as is)
- \(3! = 3 \times 2 \times 1 = 6\)

3. **Substitute back into the equation**:
\[ \frac{7 \times 6 \times 5 \times 4!}{4! \times 6} \]

4. **Cancel out \(4!\)**:
\[ \frac{7 \times 6 \times 5}{6} = 7 \times 5 = 35 \]

### (iii) When \(n = 15\) and \(r = 12\)

1. **Substitute the values into the formula**:
\[ \frac{15!}{12!(15-12)!} = \frac{15!}{12! \cdot 3!} \]

2. **Calculate the factorials**:
- \(15! = 15 \times 14 \times 13 \times 12!\)
- \(12! = 12!\) (remains as is)
- \(3! = 3 \times 2 \times 1 = 6\)

3. **Substitute back into the equation**:
\[ \frac{15 \times 14 \times 13 \times 12!}{12! \times 6} \]

4. **Cancel out \(12!\)**:
\[ \frac{15 \times 14 \times 13}{6} \]

5. **Calculate the result**:
- First, calculate \(15 \times 14 = 210\)
- Then, \(210 \times 13 = 2730\)
- Finally, divide by \(6\):
\[ \frac{2730}{6} = 455 \]

### Final Answers:
- (i) \(15\)
- (ii) \(35\)
- (iii) \(455\)