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Difficulty: HardCombinations

A research committee of 55 members is to be formed from 66 male scientists and 55 female scientists. The committee must contain strictly more male scientists than female scientists. If one particular male scientist and one particular female scientist refuse to serve on the same committee together, in how many different ways can the committee be selected?

  1. 231Answer
  2. B
    281
  3. C
    1,281
  4. D
    256

Answer

The total number of valid ways to select the committee is 231.
The correct response of 231 is found by calculating the total committees having strictly more men than women (281) and subtracting the subset of those committees where both restricted individuals are chosen together (50).

Step-by-Step Solution

1
Calculate the unconstrained ways to form a 5-member committee with strictly more males than females.
Three possible gender distributions exist:
- 5 Males, 0 Females: (65)×(50)=6×1=6\binom{6}{5} \times \binom{5}{0} = 6 \times 1 = 6
- 4 Males, 1 Female: (64)×(51)=15×5=75\binom{6}{4} \times \binom{5}{1} = 15 \times 5 = 75
- 3 Males, 2 Females: (63)×(52)=20×10=200\binom{6}{3} \times \binom{5}{2} = 20 \times 10 = 200
Total unconstrained ways = 6+75+200=2816 + 75 + 200 = 281.
Establishing the total pool of valid gender-ratio committees before applying the exclusion condition.
2
Calculate the number of forbidden committee selections where both the restricted male scientist and restricted female scientist are included together.
Since 1 male and 1 female are already chosen, 3 remaining members must be selected from the remaining 5 males and 4 females:
- For 4 Males, 1 Female total (need 3 more males, 0 more females): (53)×(40)=10×1=10\binom{5}{3} \times \binom{4}{0} = 10 \times 1 = 10
- For 3 Males, 2 Females total (need 2 more males, 1 more female): (52)×(41)=10×4=40\binom{5}{2} \times \binom{4}{1} = 10 \times 4 = 40
Total forbidden ways = 10+40=5010 + 40 = 50.
To enforce the restriction that both specific members cannot serve together, calculate all committee configurations containing both.
3
Subtract the forbidden selections from the unconstrained selections.
Valid ways = 281−50=231281 - 50 = 231.
Applying the subtraction principle of combinations to yield the final allowed arrangements.

Key Concept

Combinations with gender ratio constraints and mutual exclusion restrictions
Estimated Time:2m 30s
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Topics

Combinations

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