Question

At an election, a voter may vote for any number of candidates not exceeding the number to be elected. If 4 candidates are to be elected out of the 12 contested in the election and voter votes for at least one candidate, then the number of ways of selections is:

• A. 793
• B. 298
• C. 781
• D. 1585

Hint:

For combinations, use the formula \( ^nC_r = \frac{n!}{r!(n-r)!} \) to calculate the number of ways to choose \( r \) objects from \( n \) objects.

Answer 1

The Correct Option is A

To solve this problem, we need to determine the number of ways to select one or more candidates from a total of 12 candidates, when up to 4 can be chosen. We start by considering the cases where we select 1, 2, 3, or 4 candidates:

1. Number of ways to select 1 candidate: C(12,1)=12

2. Number of ways to select 2 candidates: C(12,2)=66

3. Number of ways to select 3 candidates: C(12,3)=220

4. Number of ways to select 4 candidates: C(12,4)=495

Adding these values gives the total number of ways:

12+66+220+495=793

Hence, the number of ways voters can choose at least one candidate is 793.

Answer 2

To determine the number of ways a voter can vote for any number of candidates (at least one, but no more than four) out of 

• Add these up: \(792+924+792+495+220+66+12+1 = 3302\).
• Subtract this sum from the total possible selections to exclude sets with more than 4 candidates: \(4095 - 3302 = 793\).

Thus, the number of ways to select at least one but no more than four candidates is 793.