Let P(n) be the statement: `C_(r)len!` for `1le r le n`
Is P(3) true ?

Answer

To determine whether the statement \( P(3) \) is true, we need to analyze the inequality given by \( C(n, r) \leq n! \) for \( 1 \leq r \leq n \) when \( n = 3 \).

### Step-by-Step Solution:

1. **Define the Statement**:
We need to check if \( C(3, r) \leq 3! \) for all \( r \) such that \( 1 \leq r \leq 3 \).

2. **Calculate \( 3! \)**:
\[ 3! = 3 \times 2 \times 1 = 6 \]

3. **Calculate \( C(3, r) \)** for \( r = 1, 2, 3 \):
- For \( r = 1 \):
\[ C(3, 1) = \frac{3!}{(3-1)! \cdot 1!} = \frac{3!}{2! \cdot 1!} = \frac{6}{2 \cdot 1} = 3 \]
- For \( r = 2 \):
\[ C(3, 2) = \frac{3!}{(3-2)! \cdot 2!} = \frac{3!}{1! \cdot 2!} = \frac{6}{1 \cdot 2} = 3 \]
- For \( r = 3 \):
\[ C(3, 3) = \frac{3!}{(3-3)! \cdot 3!} = \frac{3!}{0! \cdot 3!} = \frac{6}{1 \cdot 6} = 1 \]

4. **Check the Inequality**:
- For \( r = 1 \):
\[ C(3, 1) = 3 \leq 6 \quad \text{(True)} \]
- For \( r = 2 \):
\[ C(3, 2) = 3 \leq 6 \quad \text{(True)} \]
- For \( r = 3 \):
\[ C(3, 3) = 1 \leq 6 \quad \text{(True)} \]

5. **Conclusion**:
Since \( C(3, r) \leq 3! \) holds true for all \( r \) in the range \( 1 \leq r \leq 3 \), we conclude that \( P(3) \) is true.