If P(32, 6) = kC (32, 6), then what is the value of k?

Options

A. 6
B. 32
C. 120
D. 720

Answer

To solve the problem, we need to find the value of \( k \) such that:

\[ P(32, 6) = k \cdot C(32, 6) \]

### Step-by-Step Solution:

1. **Understand the Definitions**:
- The permutation \( P(n, r) \) is given by the formula:
\[ P(n, r) = \frac{n!}{(n-r)!} \]
- The combination \( C(n, r) \) is given by the formula:
\[ C(n, r) = \frac{n!}{r!(n-r)!} \]

2. **Substitute the Values**:
- For our problem, we have \( n = 32 \) and \( r = 6 \).
- Therefore, we can write:
\[ P(32, 6) = \frac{32!}{(32-6)!} = \frac{32!}{26!} \]
- And:
\[ C(32, 6) = \frac{32!}{6!(32-6)!} = \frac{32!}{6! \cdot 26!} \]

3. **Set Up the Equation**:
- Now, substituting these into our original equation:
\[ \frac{32!}{26!} = k \cdot \frac{32!}{6! \cdot 26!} \]

4. **Cancel Out Common Terms**:
- We can cancel \( 32! \) and \( 26! \) from both sides:
\[ 1 = k \cdot \frac{1}{6!} \]

5. **Solve for \( k \)**:
- Rearranging gives us:
\[ k = 6! \]
- Now, we calculate \( 6! \):
\[ 6! = 6 \times 5 \times 4 \times 3 \times 2 \times 1 = 720 \]

6. **Final Answer**:
- Thus, the value of \( k \) is:
\[ k = 720 \]

### Conclusion:
The value of \( k \) is \( 720 \).