Examkin

Question

Difficulty: HardPermutations, Combinations, and Fundamental Counting Principle

A city planning board needs to form a 66-member advisory task force selected from a pool of 55 architects and 55 civil engineers. The task force must include at least 22 architects and at least 22 civil engineers. However, two specific architects, Architect X and Architect Y, cannot both serve on the task force together. How many different 66-member task forces can be formed satisfying these conditions?

Answer: 135

Answer

The total number of different valid 6-member task forces that can be formed is 135.
To solve this problem, we apply the addition principle over mutually exclusive cases of committee composition, followed by complementary counting to enforce the exclusion restriction. First, we identify the valid breakdown of architects and engineers for a 6-member team requiring at least 2 of each profession: (4 architects, 2 engineers), (3 architects, 3 engineers), and (2 architects, 4 engineers). Calculating the combinations for each breakdown yields 50, 100, and 50 ways respectively, totaling 200 unconstrained team options. Next, we determine how many of these teams include both Architect X and Architect Y. Pre-assigning both architects reduces the remaining available architects to 3. The invalid cases for each breakdown are 30, 30, and 5 respectively, totaling 65 invalid configurations. Subtracting the 65 invalid teams from the 200 total unconstrained teams yields 135 valid task forces.

Step-by-Step Solution

1
Identify the allowed group breakdowns under the restriction of at least 2 architects and at least 2 engineers.
The valid (architect, engineer) count pairs for a 6-member task force are (4, 2), (3, 3), and (2, 4).
Choosing 5 architects would leave only 1 engineer, violating the minimum requirement of 2 engineers, and vice versa.
2
Compute the total combinations without the exclusion restriction.
Total unconstrained combinations = 200.
(54)(52)+(53)(53)+(52)(54)=(5×10)+(10×10)+(10×5)=50+100+50=200\binom{5}{4}\binom{5}{2} + \binom{5}{3}\binom{5}{3} + \binom{5}{2}\binom{5}{4} = (5 \times 10) + (10 \times 10) + (10 \times 5) = 50 + 100 + 50 = 200.
3
Calculate the number of task forces that violate the restriction by including both Architect X and Architect Y.
Total invalid combinations = 65.
If Architect X and Architect Y are both included, selecting remaining architects from the other 3 yields: (32)(52)+(31)(53)+(30)(54)=(3×10)+(3×10)+(1×5)=30+30+5=65\binom{3}{2}\binom{5}{2} + \binom{3}{1}\binom{5}{3} + \binom{3}{0}\binom{5}{4} = (3 \times 10) + (3 \times 10) + (1 \times 5) = 30 + 30 + 5 = 65.
4
Subtract the invalid combinations from the total unconstrained combinations.
200 - 65 = 135.
Using the complementary counting principle provides the exact number of valid combinations where Architect X and Architect Y do not serve together.

Key Concept

Combinations with multiple category constraints and complementary counting for exclusion rules.
Rate this question

Topics

Permutations, Combinations, and Fundamental Counting Principle

More questions from this topic