Question

To fill 12 vacancies, there are 25 candidates of which five are from the scheduled caste. If 3 of the vacancies are reserved for scheduled caste candidates while the rest are open to all, then the number of ways in which the selection can be made is:

• A. \( 5C3 \times 22C9 \)
• B. \( 22C9 - 5C3 \)
• C. \( 22C3 + 5C3 \)
• D. None of these

Hint:

When selecting candidates with restrictions (e.g., reserved categories), first handle the restricted choices, then handle the remaining free choices.

Answer 1

The Correct Option is A

Step 1: There are 12 vacancies to be filled, and 3 of these vacancies are reserved for candidates from the scheduled caste.
- The number of ways to select 3 candidates from the 5 scheduled caste candidates is \( 5C3 \).
- After selecting the 3 scheduled caste candidates, 9 vacancies remain, and they can be filled by any of the remaining 22 candidates. The number of ways to select 9 candidates from the 22 is \( 22C9 \).
Step 2: Therefore, the total number of ways to fill the vacancies is: \[ 5C3 \times 22C9. \]