Question 1164030: A certain club has $50$ people, and $4$ members are running for president. Each club member votes for one of the $4$ candidates. How many different possible vote totals are there?
Found 2 solutions by ikleyn, greenestamps:
Answer by ikleyn(52805)   (Show Source):
You can put this solution on YOUR website! .
Every one of 50 club members, can vote for any of 4 candidates.
The total number of all possible different votes is, therefore,  .
To make this result more clear, you may think this way.
You write the protocols of voting.
In the row of 50 positions, you write the first letters of the names of 4 candidates A, B, C or D, in each position.
Thus you get the "word" of the length of 50 letters, written in this 4-symbols alphabet {A,B,C,D}.
In the 1-st position, you may have any of 4 letters;
In the 2-nd position, you may have any of 4 letters, again, independently;
In the 3-rd position, you may have any of 4 letters, again, independently;
. . . and so on till 50-th position.
In this way, you get a huge set of all possible different protocols - all possible voting results.
The problem asks about the number of all such possible protocols,
which is the number of all possible 50-letter words, written in 4-symbol alphabet {A,B,C,D}.
In this way, you get the number , which is your ANSWER
Solved and explained.
Answer by greenestamps(13200)  (Show Source):
You can put this solution on YOUR website!
The question "how many vote totals are there" is ambiguous.
With one interpretation, since there are 50 people voting, there is only one "vote total" possible: 50.
With the interpretation from the other tutor who has responded, the number of possible vote totals is , since each of the 50 people has 4 choices of whom to vote for.
I have a different interpretation: the number of ways the 50 votes can be distributed among the 4 candidates.
Using "stars and bars" (search this site, or the internet in general, if you aren't familiar with that), the number of ways is
 = 23426}
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