# How many possible ways to vote?

## There are 8 judges. People can vote for 0, 1, 2, 3, or 4 people. How many ways can a person vote?

May 21, 2018

163 ways.

#### Explanation:

There is $1$ way to vote for 0 people.

There are $8$ ways to vote for 1 person.

There are $\frac{8 \cdot 7}{2}$ ways to vote for 2 people.

There are $\frac{8 \cdot 7 \cdot 6}{2 \cdot 3}$ ways to vote for 3 people.

There are $\frac{8 \cdot 7 \cdot 6 \cdot 5}{2 \cdot 3 \cdot 4}$ ways to vote for 4 people.

This is all because you can choose people but there are ways you can order the people. For example, there are $2 \cdot 3$ ways to order the same 3 people.

Adding everything, we get $1 + 8 + 28 + 56 + 70 = 163$.