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Difficulty: MediumCombinations and Group Selections

A committee of 44 delegates is to be selected from a group of 55 diplomats from Country X and 44 diplomats from Country Y. If 11 specific diplomat from Country X and 11 specific diplomat from Country Y cannot both serve on the committee together, how many different 44-member committees can be formed?

  1. A
    70
  2. B
    84
  3. C
    91
  4. 105Answer
  5. E
    126

Answer

105
To find the number of valid 4-member committees, calculate the total combinations of choosing 4 delegates from 9 diplomats, (94)=126\binom{9}{4} = 126. Then subtract the restricted outcomes where both specific diplomats are selected. If both specific diplomats are included, the remaining 2 positions on the committee must be chosen from the remaining 7 diplomats, which can be done in (72)=21\binom{7}{2} = 21 ways. Subtracting these invalid outcomes gives 126−21=105126 - 21 = 105.

Step-by-Step Solution

1
Calculate the total number of ways to choose 4 delegates from the total pool of 9 diplomats without any restrictions.
\binom{9}{4} = \frac{9 \times 8 \times 7 \times 6}{4 \times 3 \times 2 \times 1} = 126
The total pool consists of 5 diplomats from Country X + 4 diplomats from Country Y = 9 diplomats.
2
Calculate the number of invalid committee selections where both specific diplomats are selected together.
\binom{7}{2} = \frac{7 \times 6}{2 \times 1} = 21
If the 2 specific diplomats are both placed on the committee, 2 remaining seats must be filled from the remaining 7 diplomats.
3
Subtract the invalid combinations from the total combinations to find the number of valid committees.
126 - 21 = 105
Complementary counting gives the number of committees where both specific diplomats do not serve together.

Key Concept

Combinations with restrictions using complementary counting
Estimated Time:2m 0s
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Topics

Combinations and Group Selections

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