A committee has 11 members, 5 men and 6 women. Three different tasks are to be assigned to individual members of the committee. In how many ways can the assignments be made? In how many ways can the tasks be assigned so that both men and women are given assignments?
Answer
Combinatorics of Assigning Tasks to a Mixed Committee of Men and Women
Understanding the Scenario:
We are given a committee of 11 total members, including 5 men and 6 women. The task is to assign 3 different tasks to 3 individual members — meaning no person can receive more than one task, and the tasks themselves are distinct.
Step 1: Total Ways to Assign 3 Different Tasks to 3 Individuals
We are choosing and assigning 3 people out of 11 to 3 different tasks, which is a case of permutations:
Total ways = P(11, 3) = 11 × 10 × 9 = 990
This represents all possible ways to assign the 3 tasks to any of the 11 members without any restrictions.
Step 2: Task Assignments Where Both Genders Are Represented
We now consider only the cases where at least one man and one woman are assigned tasks. That is, we want to exclude the assignments where all 3 tasks go to only men or only women.
Case 1: All 3 Tasks Go to Men
Choose and assign 3 out of 5 men to the tasks:
P(5, 3) = 5 × 4 × 3 = 60
Case 2: All 3 Tasks Go to Women
Choose and assign 3 out of 6 women to the tasks:
P(6, 3) = 6 × 5 × 4 = 120
Valid Mixed-Gender Assignments
Subtract the above two cases from the total to get the number of task assignments where both men and women are represented:
Mixed-gender assignments = 990 − (60 + 120) = 810
Final Summary
- Total ways to assign the 3 tasks to any 3 of the 11 committee members: 990
- Ways to assign tasks ensuring both men and women are included: 810
Real-Life Application
Understanding combinations with gender or categorical constraints is useful in organizational decision-making, ensuring diverse representation in teams or assignments. This approach also applies to academic selection panels, board nominations, and inclusive hiring practices.
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