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Difficulty: MediumPermutations, Combinations, and Fundamental Counting Principle

A committee of 55 members is to be selected from a pool of 66 doctors and 44 nurses. How many different 55-member committees can be formed that contain at least 33 doctors?

Answer: 186

Answer

186
To form a 5-member committee containing at least 3 doctors from 6 doctors and 4 nurses, consider the three mutually exclusive possibilities: 3 doctors and 2 nurses, 4 doctors and 1 nurse, or 5 doctors and 0 nurses. Using combinations, the number of ways for each case are 120, 60, and 6 respectively. Summing these gives 186 distinct committees.

Step-by-Step Solution

1
Determine all valid committee compositions meeting the requirement
The committee can consist of: 3 doctors and 2 nurses, 4 doctors and 1 nurse, or 5 doctors and 0 nurses.
The prompt specifies 'at least 3 doctors' out of 5 total members.
2
Calculate the combinations for each scenario
Case 1: \(\binom{6}{3} \times \binom{4}{2} = 20 \times 6 = 120\)
Case 2: \(\binom{6}{4} \times \binom{4}{1} = 15 \times 4 = 60\)
Case 3: \(\binom{6}{5} \times \binom{4}{0} = 6 \times 1 = 6\)
Order of selection does not matter, so combination formula \(\binom{n}{k}\) is used.
3
Sum the valid combinations
120 + 60 + 6 = 186
The scenarios are mutually exclusive, so the addition principle applies.

Key Concept

Combinations with restrictions and Addition Principle
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Topics

Permutations, Combinations, and Fundamental Counting Principle

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