Question

There are 8 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?

Answer 1

8 people are running for a judge spot. Let's consider them on a case by case basis.

Case 1: 0 judges
For anything with choosing 0 objects from a total of n objects, there is only going to be one way. Thus, there is only one way to choose no judges:
$^{8}C_0 = 1$

Case 2: 1 judge
Since we want to choose just one judge, we're going to use the notation C(n, r), which means taking r objects at a time from a sample of n distinct objects.

$C(8, 1) = ^{8}C_1 = \frac{8!}{1!(8 - 1)!} = 8$

Case 3: 2 judges
Using this same logic, we can find how many ways to choose 2, 3, and 4 judges.

$\text{2 judges: } ^{8}C_2 = 28$
$\text{3 judges: } ^{8}C_3 = 56$
$\text{4 judges: } ^{8}C_4 = 70$

Adding all of these up, we get:
$\text{Total ways: } 1 + 8 + 28 + 56 + 70 = 163$