Question

There are 6 people on the ballot for 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?
SHOW ALL WORK!

Answer 1

To find the number of different ways a person can vote for the judges, we can use the combination formula.

The formula for combination is C(n, k) = n! / (k!(n-k)!) where n is the total number of choices and k is the number of choices to be made.

In this case, we have 6 choices for judges and we can vote for any 4. So n = 6 and k = 0, 1, 2, 3, or 4.

First, let's find the number of ways a person can vote for 0 judges:
C(6, 0) = 6! / (0!(6-0)!) = 6! / (0!6!) = 1

Next, let's find the number of ways a person can vote for 1 judge:
C(6, 1) = 6! / (1!(6-1)!) = 6! / 1!5! = 6

Now, let's find the number of ways a person can vote for 2 judges:
C(6, 2) = 6! / (2!(6-2)!) = 6! / 2!4! = 15

Then, let's find the number of ways a person can vote for 3 judges:
C(6, 3) = 6! / (3!(6-3)!) = 6! / 3!3! = 20

Finally, let's find the number of ways a person can vote for 4 judges:
C(6, 4) = 6! / (4!(6-4)!) = 6! / 4!2! = 15

Add up the number of ways for each case:
1 (voting for 0) + 6 (voting for 1) + 15 (voting for 2) + 20 (voting for 3) + 15 (voting for 4) = 57

Therefore, a person can vote in 57 different ways.