Question

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

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Corbin

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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.

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