Question

Five candidates are contesting an election, and three members are to be elected. A voter can vote for any number of candidates, but not more than the number of members to be elected. The number of ways a person can cast their vote is :

• A. \( 5 \)
• B. \( 15 \)
• C. \( 20 \)
• D. \( 25 \) Correct Answer: (D) \( 25 \) Solution:

Hint:

Always read carefully if the question allows for a "zero" vote (abstaining). If it did, the answer would be \( \sum_{r=0}^{3} ^5C_r = 1 + 25 = 26 \).
Also, for small values like \( n=5 \), sketching the Pascal's triangle row \( (1, 5, 10, 10, 5, 1) \) is a very fast way to get the values for \( ^5C_r \).

Answer 1

The Correct Option is D

Step 1: Understanding the Concept:
This is a counting problem involving combinations. In an election where multiple seats are available, a voter usually has the freedom to choose how many people they want to support, up to the limit of available seats.
The phrase "not more than the number of members to be elected" establishes an upper bound. Since 3 members are to be elected, a voter can choose to vote for 1 person, 2 people, or 3 people.
Note: In standard combinatorial problems of this type, we assume the voter must vote for at least one person (casting a blank ballot is usually not counted unless specified). Since the options do not include 26 (\( 25 + 1 \)), we exclude the 0-vote case.

Step 2: Key Formula or Approach:
We use the combination formula \( ^nC_r \), which calculates the number of ways to select \( r \) items from a set of \( n \) distinct items where order doesn't matter.
\[ ^nC_r = \frac{n!}{r!(n-r)!} \]
The total number of ways is the sum of the mutually exclusive scenarios:
Total = (Ways to choose 1) + (Ways to choose 2) + (Ways to choose 3).

Step 3: Detailed Explanation:
We have \( n = 5 \) candidates. The voter can choose \( r \in \{1, 2, 3\} \).
Step 3.1: Calculate \( ^5C_1 \).
The number of ways to pick exactly one candidate from five:
\[ ^5C_1 = 5 \]
Step 3.2: Calculate \( ^5C_2 \).
The number of ways to pick exactly two candidates from five:
\[ ^5C_2 = \frac{5 \cdot 4}{2 \cdot 1} = 10 \]
Step 3.3: Calculate \( ^5C_3 \).
The number of ways to pick exactly three candidates from five. Using the property \( ^nC_r = ^nC_{n-r} \):
\[ ^5C_3 = ^5C_2 = 10 \]
Step 3.4: Summing the totals.
Total ways = \( 5 \text{ (one vote)} + 10 \text{ (two votes)} + 10 \text{ (three votes)} \)
Total ways = \( 25 \).
Comparing this result with the options, it matches Option (D).

Step 4: Final Answer:
By considering all valid voting scenarios (voting for 1, 2, or 3 candidates) and summing the possible combinations for each, we find that there are 25 distinct ways for a voter to cast their vote. Therefore, Option (D) is the correct choice.