Question

In a college election, 5 candidates contested and 100 students voted. If each student voted for 2 candidates, and each pair of candidates received the same number of votes, how many votes did each candidate get?

• A. 40
• B. 50
• C. 60
• D. 80

Hint:

In double-counting setups, count votes per pair, then distribute based on how many times each candidate appears in such pairs.

Answer 1

The Correct Option is C

Each student votes for 2 candidates → total votes = \(100 \times 2 = 200\) votes Let’s count votes from candidate’s perspective. If all pairs of candidates received equal votes, we calculate how many such pairs exist: \[ \text{Number of unique pairs of 5 candidates} = \binom{5}{2} = 10 \Rightarrow \text{Each pair received } \frac{200}{10} = 20 \text{ votes} \] Each candidate is part of \( \binom{4}{1} = 4 \) pairs (with other 4 candidates) \[ \text{So each candidate appears in 4 pairs × 20 votes = 80 votes} \text{But since each vote includes two candidates, each candidate is counted in two pairs per vote} \Rightarrow \text{Actual votes per candidate} = \frac{80}{2} = \boxed{60} \]