An election ballot asks voters to select three city judges from a group of 12 candidates. How many ways can this be done?

This problem involves combinations, as the order in which the judges are selected does not matter. The formula for combinations is given by: \[ C(n, k) = \frac{n!}{k!(n-k)!} \] where \(n!\) denotes the factorial of \(n\), which is the product of all positive integers up to \(n\). In this case, there are 12 candidates, and voters need to select 3 judges. Therefore, the number of ways to select 3 judges from a group of 12 candidates is given by: \[ C(12, 3) = \frac{12!}{3!(12-3)!} \] Let's calculate this: \[ C(12, 3) = \frac{12!}{3! \cdot 9!} \] \[ C(12, 3) = \frac{12 \cdot 11 \cdot 10}{3 \cdot 2 \cdot 1} \] \[ C(12, 3) = 220 \] So, there are 220 ways to select three city judges from a group of 12 candidates.