Forming a Committee: A Comprehensive Guide With Constraints
Forming a Committee: A Comprehensive Guide With Constraints
Committees are a common feature in many organizational settings, requiring careful selection based on specific criteria. In this article, we explore a particular challenge in committee formation, involving constraints such as one girl refusing to serve and a certain girl being chosen. We'll delve into the combinatorial mathematics behind the process and provide a step-by-step solution.
Problem Statement and Constraints
The problem scenario involves forming a committee consisting of 3 boys and 4 girls from a pool of 5 boys and 6 girls. There are two key constraints:
One girl refuses to serve. A certain girl must be chosen for the committee.Understanding these constraints is crucial for calculating the number of possible committees.
Step-by-Step Solution
Step 1: Identify the Relevant Groups
For the given constraints, we start by identifying the relevant groups:
Total boys: 5 Total girls: 6 Girls available after excluding the one who refuses: 5 (since 1 girl refuses)Step 2: Choose the Certain Girl
Let's denote the certain girl who must be chosen as Girl A. Since she is selected, we need to choose 3 additional girls from the remaining 4 girls:
text{Number of ways to choose 3 girls from 4} binom{4}{3} 4
Step 3: Choose the Boys
Next, we need to choose 3 boys from the 5 available boys:
text{Number of ways to choose 3 boys from 5} binom{5}{3} 10
Step 4: Calculate the Total Number of Committees
To find the total number of possible committees, we multiply the number of ways to choose the girls by the number of ways to choose the boys:
text{Total committees} binom{4}{3} times binom{5}{3} 4 times 10 40
Conclusion
Thus, the total number of committees possible under the given conditions is 40.
Breakdown of the Problem
Let's further break down the problem into two parts:
Part 1: The Number of Committees Without Any Restrictions
You need to choose 3 boys from 5, which can be done in:
C53 cdot C64 10 cdot 15 150.
Part 2: The Number of Committees When One Girl Refuses to Serve
Let's call the girls G1, G2, G3, G4, G5, and G6. If G1 refuses to serve with G2, consider the following two cases:
G1 is in the committee and G2 is not. You need to choose 3 boys from the remaining 5 boys and 3 girls from the remaining 5 girls excluding G2. This can be done in: C53 cdot C53 10 cdot 10 100. G2 is in the committee and G1 is not. This is the same as the first case, so it is also 100.So the total number of committees with the given restriction is 100 100 200.
Therefore, there are 150 committees possible without any restrictions and 200 committees possible when one girl refuses to serve if a certain other girl is chosen.
Key Takeaways
In committee formation, it's crucial to understand the constraints and use combinatorial methods to calculate the possible outcomes. This article provides a detailed breakdown of how to approach similar problems, ensuring a deep understanding of the mathematical principles involved.