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…

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.