Select Candidates based on votes count permutations

Question

I have a question i am not sure which formulae fit for such scenario

Twenty people are voting for one of 5 candidates. They have secret ballot, each voter votes for one of 5 candidates. The result of an election is the number of votes for each of the candidate. How many possible results can this vote have?

Answer 1

Remember that the following are equivalent:

• The number of integer solutions of $$x_1+x_2+...+x_n=r, \ \ \ \ \ x_i\geq0$$
• Number of selections, with repetition, of size $r$ from a collections of size $n$.
• Number of ways $r$ identical objects can be distributed among $n$ distinct containers.

Let each candidate be $x_i$ for $i \in \{1,2,3,4,5 \}$. Then you are looking for the number of nonnegative integer solutions for $$x_1+x_2+x_3+x_4+x_5 = 20.$$ This is equivalent to finding the number of selections, with repetition, of size $20$ from a collection of size $5$, which is given by $$\binom{5+20-1}{20} = \binom{24}{20} = 10,626.$$ So you have $10,626$ possible results for the election. This is because you are not taking into account the identity of the voters, so the $20$ votes are counted as identical objects to be distributed among $5$ distinct "containers", which, in this case, would refer to the candidates.