At election there are 5 candidates and 3 members are to be elected and a voter is entitled to vote for any number to be elected. In how many ways a voter can a vote ?

Options

A. 25
B. `5^3`
C. `3^5`
D. 15

Answer

To solve the problem of how many ways a voter can vote in an election with 5 candidates where 3 members are to be elected, we can break it down step by step.

### Step 1: Understand the Voting Options
The voter can vote for any number of candidates from 0 to 3. This means the voter can choose:
- 0 candidates (not voting)
- 1 candidate
- 2 candidates
- 3 candidates

### Step 2: Calculate the Number of Ways to Vote for Each Case
1. **Choosing 0 candidates**: There is only 1 way to do this (not voting).
- Ways = 1

2. **Choosing 1 candidate**: The voter can choose any 1 out of the 5 candidates. The number of ways to choose 1 candidate from 5 is given by the combination formula \( \binom{n}{r} \), where \( n \) is the total number of candidates and \( r \) is the number of candidates to choose.
- Ways = \( \binom{5}{1} = 5 \)

3. **Choosing 2 candidates**: The number of ways to choose 2 candidates from 5 is:
- Ways = \( \binom{5}{2} = 10 \)

4. **Choosing 3 candidates**: The number of ways to choose 3 candidates from 5 is:
- Ways = \( \binom{5}{3} = 10 \)

### Step 3: Sum Up All the Ways
Now, we add up all the ways a voter can vote:
- Ways to choose 0 candidates: 1
- Ways to choose 1 candidate: 5
- Ways to choose 2 candidates: 10
- Ways to choose 3 candidates: 10

Total ways = \( 1 + 5 + 10 + 10 = 26 \)

### Final Answer
The total number of ways a voter can vote is **26**.

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