A selection is to be made for one post of Principal and two posts of Vice-Principal. Amongst the six candidates called for the interview, only two are eligible for the post of Principal while they all are eligible for the post of Vice-Principal. The number of possible combinations of selectees is
- (a) 4
- (b) 12
- (c) 18
- (d) None of the above ✓ UPSC's answer
Why the answer is (d)
• There are 6 candidates in total, with only 2 eligible for the Principal post and all 6 eligible for the Vice-Principal post.
• First, select the Principal from the 2 eligible candidates, which can be done in $^2C_1 = 2$ ways.
• Next, select 2 Vice-Principals from the remaining 5 candidates (since the Principal is already selected and cannot hold both posts), which can be done in $^5C_2 = 10$ ways.
• The total number of possible combinations is the product of these two steps: $2 \times 10 = 20$.
• Since 20 is not listed in options (a), (b), or (c), the correct choice is (d) None of the above.
Why the other options are wrong
- (a) 4
- Option (a) is incorrect because the total number of combinations is 20, not 4.
- (b) 12
- Option (b) is incorrect because the total number of combinations is 20, not 12.
- (c) 18
- Option (c) is incorrect because the total number of combinations is 20, not 18.
Asked in the CSAT Paper II of the UPSC Civil Services Preliminary Examination 2015, held on 23 August 2015. Question and answer key: Union Public Service Commission. Explanation: UPSC Answer Check.