Calculating the Number of Committees: A Comprehensive Guide
When forming a committee with specific roles, such as two teachers and three students from a pool of 5 teachers and 50 students, we can use combinatorial methods. This article walks you through the process step-by-step, utilizing mathematical principles to achieve an accurate count. We'll explore combinations and provide a thorough example to ensure clarity.
Introduction to Combinatorics
Combinatorics is a branch of mathematics that deals with the study of discrete structures, including combinations and permutations. In this context, combinations refer to the selection of items from a larger pool without regard to order. This makes it particularly useful for problems like forming committees.
The Problem: Selecting Teachers and Students
Given the problem at hand, we need to determine how many different committees can be formed from 5 teachers and 50 students, with each committee comprising 2 teachers and 3 students. This can be solved using the formula for combinations, denoted as (binom{n}{r}), where (n) is the total number of items to choose from, and (r) is the number of items to choose. The general formula for combinations is:
(binom{n}{r} frac{n!}{r!(n-r)!})Step-by-Step Calculation
Step 1: Choosing 2 Teachers
The first step involves choosing 2 teachers from a group of 5. Applying the combination formula:
(binom{5}{2} frac{5!}{2!(5-2)!})Breaking this down:
(binom{5}{2} frac{5 times 4}{2 times 1} 10)
Thus, there are 10 different ways to choose 2 teachers from 5.
Step 2: Choosing 3 Students
The next step is to select 3 students from the pool of 50. Applying the combination formula again:
(binom{50}{3} frac{50!}{3!(50-3)!})Breaking this down:
(binom{50}{3} frac{50 times 49 times 48}{3 times 2 times 1} frac{117600}{6} 19600)
Therefore, there are 19,600 different ways to choose 3 students from 50.
Step 3: Total Number of Committees
To find the total number of possible committees, we multiply the number of ways to choose the teachers by the number of ways to choose the students:
(text{Total Committees} binom{5}{2} times binom{50}{3} 10 times 19600 196000)Thus, the total number of committees that can be formed is 196,000.
Alternative Method: Combinatorial Approach
If you prefer a different approach, the problem can be understood as follows:
We start by noting that from 5 teachers, choosing any 2 results in 20 possible combinations when considering the order of selection. However, since the order of choosing the teachers does not matter, we divide by 2 (the number of ways to arrange 2 teachers). From 50 students, choosing 3 results in a combination of (50 times 49 times 48) choices. Since the order of selection does not matter, we divide by 6 (the number of ways to arrange 3 students).Therefore, the calculation is:
(text{Total Committees} frac{20 times 50 times 49 times 48}{2 times 6} 10 times frac{50 times 49 times 48}{6} 10 times 19600 196000)Conclusion
The detailed step-by-step calculation demonstrates the application of combinatorial methods to solve the problem of forming committees. This approach is not only mathematically robust but also provides a clear understanding of the underlying principles. Whether you prefer the direct application of the combination formula or the alternative combinatorial approach, the result remains the same: there are 196,000 possible committees.
Frequently Asked Questions
Q: What are combinations?
Combinations are used to determine the number of ways to choose a subset of items from a larger set without regard to order. This is particularly useful for forming committees, selecting groups of students, or any scenario where the order of selection is irrelevant.
Q: Can you simplify the calculation?
Yes, simplifying the calculation involves understanding that the order of selection does not matter. This leads to dividing by the number of permutations of the chosen items (2 for teachers and 6 for students).
Q: How does this apply in real-world scenarios?
This method is applicable in various real-world scenarios, such as forming groups for projects, allocating tasks, and organizing teams. By using combinatorial methods, we can ensure a clear and systematic approach to problem-solving.