If `.^(6)P_(r): .^(6)P_(5) = 1:2`, find the value of r.

Explanation

To solve the problem, we need to find the value of \( r \) given the ratio \( \frac{^6P_r}{^6P_5} = \frac{1}{2} \).

### Step-by-Step Solution:

1. **Understanding the Permutation Formula**:
The formula for permutations is given by:
\[
^nP_r = \frac{n!}{(n-r)!}
...