If `.^(n)C_(5) = .^(n)C_(7)`, then find `.^(n)P_(3)`

Answer

To solve the problem where \( \binom{n}{5} = \binom{n}{7} \) and find \( P(n, 3) \), we can follow these steps:

### Step 1: Understand the condition \( \binom{n}{5} = \binom{n}{7} \)
From the properties of combinations, we know that if \( \binom{n}{r} = \binom{n}{k} \), then either:
1. \( n = r \) or \( n = k \)
2. \( n = r + k \)

In our case, we have \( r = 5 \) and \( k = 7 \). Since \( 5 \neq 7 \), we can use the second condition:
\[ n = 5 + 7 \]

### Step 2: Calculate \( n \)
Now we can calculate \( n \):
\[ n = 5 + 7 = 12 \]

### Step 3: Find \( P(n, 3) \)
Now that we have \( n = 12 \), we need to find \( P(12, 3) \). The formula for permutations is given by:
\[ P(n, r) = \frac{n!}{(n - r)!} \]
Substituting \( n = 12 \) and \( r = 3 \):
\[ P(12, 3) = \frac{12!}{(12 - 3)!} = \frac{12!}{9!} \]

### Step 4: Simplify \( P(12, 3) \)
We can simplify \( \frac{12!}{9!} \):
\[ P(12, 3) = \frac{12 \times 11 \times 10 \times 9!}{9!} \]
The \( 9! \) cancels out:
\[ P(12, 3) = 12 \times 11 \times 10 \]

### Step 5: Calculate the final value
Now we can calculate:
\[ 12 \times 11 = 132 \]
\[ 132 \times 10 = 1320 \]

Thus, the final answer is:
\[ P(12, 3) = 1320 \]

### Final Answer:
\[ \boxed{1320} \]