Step 1: The combination matching rule.
If $^{n}C_x = {}^{n}C_y$ and $x \ne y$, then the two lower numbers must add up to $n$, that is $x + y = n$.
Step 2: Find $n$.
Here $x = 12$ and $y = 8$, which are different, so \[ n = 12 + 8 = 20 \]
Step 3: Rewrite the target.
We need $^{20}C_{17}$. Using $^{n}C_r = {}^{n}C_{n-r}$, this equals $^{20}C_3$, which is easier.
Step 4: Write the formula.
\[ ^{20}C_3 = \frac{20 \times 19 \times 18}{3 \times 2 \times 1} \]
Step 5: Cancel before multiplying.
$18 \div 6 = 3$, so the top becomes $20 \times 19 \times 3$.
Step 6: Multiply.
\[ 20 \times 19 \times 3 = 1140 \] \[ \boxed{ ^{n}C_{17} = 1140 } \]