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

Answer

To solve the problem where \(\binom{n}{5} = \binom{n}{7}\), we need to find the value of \(P(n, 3)\).

### Step-by-Step Solution:

1. **Understanding the Equality of Combinations**:
We know that \(\binom{n}{r} = \binom{n}{n-r}\). This means that \(\binom{n}{5} = \binom{n}{7}\) implies:
\[ 5 = n - 7 \]

2. **Solving for \(n\)**:
Rearranging the equation gives:
\[ n = 5 + 7 = 12 \]

3. **Finding \(P(n, 3)\)**:
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 \(r = 3\):
\[ P(12, 3) = \frac{12!}{(12-3)!} = \frac{12!}{9!} \]

4. **Simplifying the Permutation**:
We can simplify this as follows:
\[ P(12, 3) = 12 \times 11 \times 10 \times \frac{9!}{9!} = 12 \times 11 \times 10 \]

5. **Calculating the Final Value**:
Now, we calculate:
\[ 12 \times 11 = 132 \]
\[ 132 \times 10 = 1320 \]

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