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?

55,396

based on 4504 people

and 444 members running for president.

If your question was only 4 members running for president then you would just need to not thank me and allow me to edit.

We would simply show maximum if 4 by dividing 4504/4 = 1126 equal distribution and follow the pattern below using 1126 votes only as a total after multiplying it like this 1126 x 1128 x 4- 4 = 5,080,508

This is because the distribution of how many people voted the same person even if they only had 4 people to vote for makes the total votes higher than that of a larger selection of people to vote for which means the maximum vote has increased for a smaller selection. So if there are only 4 members running for president the actual votes become 5,080,508 possible votes.

If we have 444 members running for presidency, then the vote totals distributed become 55,396 in total.

Step-by-step explanation:

4504 people and 444 members are running for president

Each club member votes for one of the 444 candidates

4505 / 444 = 10.1463963964

The exact votes could be 10 each as distribution = 444 equal votes

Then we can keep applying one and loosing one = 443, 442,441, 440 etc.

This alone would equal 10 of 444 x 12 - 444 = 10 x 12 x 444 -444 = 52836

We then see a balance of distribution remainder as 10.1463963964 shows a remainder of  0.1463963964 so we divide into 1

1/ 0.1463963964 = 6.8307692306

Which means every 6 vote decrease there is another vote candidate.

444/6 = 74 extra vote that would have 1  = 74

73 extra vote that would have 1 + 1 that has 2 = 74

72 extra vote that would have 1+2 that has 3 = 73

and down to one we can calculate 74 x 74 - 74 - 73 - 72 - 71- 72- 71 - 70- 69-68

= 4836

then continue subtraction

4836 - 67-66-65-64-63-62-61-60 = 4328

4320 - 59-58-57056-55-54-53-52-51-50 = 3775

3775 - 49-48-47-46-45-44-43-42-41 -40 = 3330

3330 - 39-38-37-36-35-34-33-32-31-30 = 2985

2985 - 29-28-27-26-25-24-23-22-21-20 = 2750

2750 - 19-18-17-16-15-14-13-12-11-10 = 2605

2605 - 9-8-7-6-5-4-3-2-1 = 2560

We cna now add this total with the previous =  2560 + 52836 = 55,396

the answer is 62