Questions: An election ballot asks voters to select six city commissioners from a group of twenty candidates. In how many ways can this be done

Transcript text: An election ballot asks voters to select six city commissioners from a group of twenty candidates. In how many ways can this be done

Solution

Solution Steps

To determine the number of ways to select six city commissioners from a group of twenty candidates, we need to use the concept of combinations. The formula for combinations is given by:

\[ C(n, k) = \frac{n!}{k!(n-k)!} \]

where \( n \) is the total number of candidates, and \( k \) is the number of candidates to be selected.

Step 1: Identify the Problem

We need to determine the number of ways to select six city commissioners from a group of twenty candidates. This is a combination problem where we are choosing \( k \) items from \( n \) items without regard to the order.

Step 2: Use the Combination Formula

The formula for combinations is given by: \[ C(n, k) = \frac{n!}{k!(n-k)!} \] where \( n \) is the total number of candidates, and \( k \) is the number of candidates to be selected.

Step 3: Substitute the Values

Substitute \( n = 20 \) and \( k = 6 \) into the combination formula: \[ C(20, 6) = \frac{20!}{6!(20-6)!} = \frac{20!}{6! \cdot 14!} \]

Step 4: Calculate the Result

Using the combination formula, we calculate: \[ C(20, 6) = 38,760 \]