If `^nP_r=120^nC_r` then r is equal to-

Options

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

Answer

To solve the equation \( nP_r = 120 \cdot nC_r \), 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**:
\[ \frac{n!}{(n-r)!} = 120 \cdot \frac{n!}{r!(n-r)!} \]

3. **Cancel \( n! \) and \( (n-r)! \) from both sides** (assuming \( n! \neq 0 \) and \( (n-r)! \neq 0 \)):
\[ 1 = 120 \cdot \frac{1}{r!} \]

4. **Rearranging the equation**:
\[ r! = 120 \]

5. **Find the value of \( r \)**:
- We know that \( 120 = 5! \) (since \( 5! = 5 \times 4 \times 3 \times 2 \times 1 = 120 \)).
- Therefore, \( r = 5 \).

### Final Answer:
Thus, the value of \( r \) is \( 5 \).

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