Question

There are 6 people on the ballot for regional judges. Voters can vote for any 4. Voters can choose to vote for 0, 1, 2, 3, or 4 judges. In how many different ways can a person vote?(1 point)
Responses

5
5

15
15

6
6

57

Answer 1

To determine how many different ways a person can vote for the judges, we can consider the number of ways to choose 0, 1, 2, 3, or 4 judges from a group of 6 judges.

We use the binomial coefficient \( \binom{n}{k} \), which represents the number of ways to choose \( k \) items from \( n \) items. In this case, we will calculate \( \binom{6}{k} \) for \( k = 0, 1, 2, 3, 4 \) and then sum these values.

1. **Choosing 0 judges:**
\[
\binom{6}{0} = 1
\]

2. **Choosing 1 judge:**
\[
\binom{6}{1} = 6
\]

3. **Choosing 2 judges:**
\[
\binom{6}{2} = \frac{6 \times 5}{2 \times 1} = 15
\]

4. **Choosing 3 judges:**
\[
\binom{6}{3} = \frac{6 \times 5 \times 4}{3 \times 2 \times 1} = 20
\]

5. **Choosing 4 judges:**
\[
\binom{6}{4} = \binom{6}{2} = 15
\]

Now, we add these values together to find the total number of ways a person can vote:

\[
\text{Total} = \binom{6}{0} + \binom{6}{1} + \binom{6}{2} + \binom{6}{3} + \binom{6}{4} = 1 + 6 + 15 + 20 + 15 = 57
\]

Thus, the total number of different ways a person can vote is **57**.