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?
This problem involves combinations, since the order in which the judges are chosen does not matter. We can use the combination formula:
n C r = n! / (r! (n-r)!)
where n is the total number of judges (6) and r is the number of judges to be chosen (from 0 to 4).
To count the total number of ways a person can vote, we can sum up the combinations for r = 0, 1, 2, 3, and 4:
6 C 0 + 6 C 1 + 6 C 2 + 6 C 3 + 6 C 4
= 1 + 6 + 15 + 20 + 15
= 57
Therefore, there are 57 different ways a person can vote for the regional judges.
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To find the number of different ways a person can vote, we need to consider all possible scenarios for each judge choice: whether to vote for them or not.
Since voters can choose to vote for 0, 1, 2, 3, or 4 judges, we can calculate the total number of ways by adding up the number of ways for each possible choice. Let's break it down:
For the first judge, there are two choices: vote for them or not vote for them (1 or 0).
For the second judge, again, there are two choices: vote for them or not vote for them (1 or 0).
Similarly, the third judge and the fourth judge also have two choices each.
Therefore, the total number of ways to vote can be calculated as follows:
Total number of ways = 2 * 2 * 2 * 2
Simplifying this, we have:
Total number of ways = 2^4
Calculating 2^4, we find:
Total number of ways = 16
So, a person can vote in 16 different ways in this scenario.