If`^nP_r=720^nC_r`than the value r is-

Options

A. 6
B. 5
C. 4
D. 7

Answer

To solve the equation \( ^nP_r = 720 \times ^nC_r \), we will use the formulas for permutations and combinations.

### Step-by-Step Solution:

1. **Write the formulas for permutations and combinations:**
\[ ^nP_r = \frac{n!}{(n-r)!} \]
\[ ^nC_r = \frac{n!}{r!(n-r)!} \]

2. **Substitute the formulas into the equation:**
\[ \frac{n!}{(n-r)!} = 720 \times \frac{n!}{r!(n-r)!} \]

3. **Cancel \( n! \) and \( (n-r)! \) from both sides (assuming \( n! \neq 0 \) and \( (n-r)! \neq 0 \)):**
\[ 1 = 720 \times \frac{1}{r!} \]

4. **Rearrange the equation to isolate \( r! \):**
\[ r! = 720 \]

5. **Find the value of \( r \) by determining which factorial equals 720:**
- Calculate factorials:
- \( 1! = 1 \)
- \( 2! = 2 \)
- \( 3! = 6 \)
- \( 4! = 24 \)
- \( 5! = 120 \)
- \( 6! = 720 \)

From the calculations, we find that \( 6! = 720 \).

6. **Thus, the value of \( r \) is:**
\[ r = 6 \]

### Final Answer:
The value of \( r \) is \( 6 \).