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There are $3$ candidates and $5$ voters.
(a) In how many ways can the votes be given?
(b) If the $3$ candidates are among the $5$ voters and they vote for themselves only. Then, how many ways of giving votes are possible?

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Last updated date: 13th Jul 2024
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Answer
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Hint: The question revolves around the concepts of permutations and combinations. We are given the number of candidates and voters in an election and we are required to find the number of possible ways of giving votes. According to the fundamental theorem of counting, if there are p ways of doing one thing and q ways of doing another thing, then there are $p \times q$ ways of doing both the things. So, we find out the number of options with each voter and then multiply the number of options with each voter so as to get the required answer.

Complete step-by-step solution:
So, we are given the number of voters as $5$.
Also, the number of candidates in the election is $3$.
(a) Now, voters have to choose a candidate out of these three candidates in the elections.
So, each voter has three options to choose a candidate in the elections.
So, the number of ways in which a voter can choose a candidate in the elections is $3$.
Now, using the fundamental principle of counting, we get
Number of ways in which the votes can be given by the five voters as $3 \times 3 \times 3 \times 3 \times 3$
$ = {3^5} = 243$
Hence, the number of ways in which the votes can be given are $243$.
(b) Now, consider the second situation where the $3$ candidates are among the $5$ voters and they vote for themselves only.
Since the $3$ candidates are among the $5$ voters and they only vote for themselves. So, there is one way for the candidates to choose themselves in the elections.
Now, we will calculate the number of possibilities with the $2$ voters that are not candidates themselves. So, they can vote for any of the $3$ candidates. Hence, they have $3$ options to choose from.
So, the number of ways in which a voter that is not a candidate can choose a candidate in the elections is $3$.
Number of ways in which the votes can be given by the five voters as $1 \times 1 \times 1 \times 3 \times 3$
$ = {3^2} = 9$
Hence, there are $9$ ways of giving votes if the $3$ candidates are among the $5$ voters and they vote for themselves only.

Note: Combination means choosing elements is only that matters, whereas permutation is an ordered combination. The formula used to find combination is \[{}^n{C_r} = \dfrac{{n!}}{{r! \times \left( {n - r} \right)!}}\] . Permutation is a method used to calculate the total outcome of a situation where order is important, the formula used to find permutation is and \[{}^n{P_r} = \dfrac{{n!}}{{\left( {n - r} \right)!}}\].