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:
[tex]^{8}C_0 = 1[/tex]
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.
[tex]C(8, 1) = ^{8}C_1 = \frac{8!}{1!(8 - 1)!} = 8[/tex]
Case 3: 2 judges
Using this same logic, we can find how many ways to choose 2, 3, and 4 judges.
[tex]\text{2 judges: } ^{8}C_2 = 28[/tex]
[tex]\text{3 judges: } ^{8}C_3 = 56[/tex]
[tex]\text{4 judges: } ^{8}C_4 = 70[/tex]
Adding all of these up, we get:
[tex]\text{Total ways: } 1 + 8 + 28 + 56 + 70 = 163[/tex]
