In the club election, the number of contestants is one more than the number of maximum candidates for which a voter can vote. If the total number of ways in which a voter can vote be 62, then the number of candidates is:

Options

A. 7
B. 5
C. 6
D. 8

Answer

To solve the problem step by step, we can follow these steps:

### Step 1: Understand the Problem
We have a club election where the number of contestants is one more than the maximum number of candidates a voter can vote for. We need to find out how many candidates there are, given that the total number of ways a voter can vote is 62.

### Step 2: Define Variables
Let:
- \( N \) = the maximum number of candidates a voter can vote for.
- The total number of contestants = \( N + 1 \).

### Step 3: Calculate Total Voting Ways
The total number of ways a voter can vote for \( N + 1 \) candidates is calculated using the formula for combinations. A voter can choose to vote for 0 to \( N \) candidates from \( N + 1 \) contestants. The total number of ways to vote can be expressed as:

\[ \text{Total ways} = \sum_{k=0}^{N} \binom{N+1}{k} \]

This is equal to \( 2^{N+1} \) (the total number of subsets of \( N + 1 \) candidates).

### Step 4: Set Up the Equation
Given that the total number of ways to vote is 62, we can set up the equation:

\[ 2^{N+1} = 62 \]

### Step 5: Solve for \( N \)
To solve for \( N \), we first need to express 62 as a power of 2. However, we notice that \( 62 \) is not a power of \( 2 \). Instead, we can find the closest power of \( 2 \):

\[ 2^6 = 64 \quad \text{and} \quad 2^5 = 32 \]

Since \( 62 \) is between \( 32 \) and \( 64 \), we can find \( N + 1 \):

\[ N + 1 = 6 \implies N = 5 \]

### Step 6: Conclusion
Thus, the number of candidates is:

\[ \text{Number of candidates} = N + 1 = 5 + 1 = 6 \]

### Final Answer
The number of candidates is **6**.