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

We are given 12 contested candidates, and we need to select at least one candidate for the election.

The total number of ways to make a selection, considering at least one candidate is chosen, is calculated as:

\[ \sum_{r=1}^{4} {^{12}C_r} = {^{12}C_1} + {^{12}C_2} + {^{12}C_3} + {^{12}C_4} \]

Step 1: Calculate individual combinations

\[ ^{12}C_1 = 12, \quad ^{12}C_2 = 66, \quad ^{12}C_3 = 220, \quad ^{12}C_4 = 495. \]

Step 2: Summing up all selections

\[ 12 + 66 + 220 + 495 = 793. \]

Final Answer: The total number of ways to select candidates is 793, which matches option (A).