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

Correct Option (4)

The selection process requires choosing one Principal and two Vice-Principals from a pool of six candidates, with specific eligibility criteria.

• Selection of Principal: There are 2 candidates who are eligible for the post of Principal. The number of ways to select 1 Principal from these 2 eligible candidates is calculated using the combination formula ⁿC_r = n! / (r!(n-r)!):
  ²C₁ = 2! / (1!(2-1)!) = 2! / (1!1!) = 2.
• Selection of Vice-Principals: After one Principal has been selected, 5 candidates remain from the original 6. All these 5 remaining candidates are eligible for the post of Vice-Principal. The number of ways to select 2 Vice-Principals from these 5 candidates is:
  ⁵C₂ = 5! / (2!(5-2)!) = 5! / (2!3!) = (5 × 4 × 3!) / (2 × 1 × 3!) = (5 × 4) / 2 = 10.
• Total Combinations: The total number of possible combinations for selecting one Principal and two Vice-Principals is the product of the number of ways for each independent selection:
  Total combinations = ²C₁ × ⁵C₂ = 2 × 10 = 20.

Since the calculated number of possible combinations is 20, and this value is not provided in options 1, 2, or 3, option 4 "None of the above" is the correct answer.

Incorrect Options:

Options 1 (4), 2 (12), and 3 (18) are incorrect. These values do not correspond to the accurate calculation of possible combinations for selecting one Principal and two Vice-Principals according to the specified eligibility and selection criteria. The correct application of combinatorial principles yields 20 combinations.