There are 5 candidates in an election and 3 of them are to be elected. A voter can cast any number of votes but not more than three. The number of ways in which he can cast his vote is

Options

A. 5
B. 15
C. 20
D. 25

Answer

To solve the problem of how many ways a voter can cast votes for 5 candidates when they can vote for up to 3 candidates, we can break it down into steps:

### Step-by-Step Solution:

1. **Understanding Voting Options**:
The voter can cast votes in three different scenarios:
- Vote for 1 candidate.
- Vote for 2 candidates.
- Vote for 3 candidates.

2. **Calculating Ways to Vote for 1 Candidate**:
- The number of ways to choose 1 candidate from 5 is given by the combination formula \( C(n, r) \), where \( n \) is the total number of candidates and \( r \) is the number of candidates to choose.
- Here, it is \( C(5, 1) = \frac{5!}{(5-1)! \cdot 1!} = \frac{5!}{4! \cdot 1!} = 5 \).

3. **Calculating Ways to Vote for 2 Candidates**:
- The number of ways to choose 2 candidates from 5 is \( C(5, 2) \).
- This is calculated as \( C(5, 2) = \frac{5!}{(5-2)! \cdot 2!} = \frac{5!}{3! \cdot 2!} = \frac{5 \times 4}{2 \times 1} = 10 \).

4. **Calculating Ways to Vote for 3 Candidates**:
- The number of ways to choose 3 candidates from 5 is \( C(5, 3) \).
- This is calculated as \( C(5, 3) = \frac{5!}{(5-3)! \cdot 3!} = \frac{5!}{2! \cdot 3!} = \frac{5 \times 4}{2 \times 1} = 10 \).

5. **Total Ways to Vote**:
- Now, we sum the number of ways from each scenario:
\[ \text{Total Ways} = C(5, 1) + C(5, 2) + C(5, 3) = 5 + 10 + 10 = 25. \]

### Final Answer:
The total number of ways a voter can cast their vote is **25**.

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