Forming a Committee with Specific Senior and Junior Members
When faced with the task of forming a committee of 10 members, with a specific requirement of having exactly 6 seniors from a pool of 17 seniors and 15 juniors, the question of how many ways this can be done becomes a matter of mathematical interest. This article will explore the combination formula and provide a detailed explanation of how to calculate the number of possible committees under these conditions.
Calculation of Possible Committees
First, let's consider the selection of 6 seniors from the 17 available seniors. This can be calculated using the combination formula binom{n}{k}, which is used to find the number of ways to choose k items from n items without regard to order.
The required combination for 6 seniors from 17 is given by:
[binom{17}{6} frac{17!}{6!(17-6)!} 12376]Next, the remaining 4 members of the committee must be juniors selected from the 15 available juniors. Again, this can be calculated using the combination formula:
[binom{15}{4} frac{15!}{4!(15-4)!} 1365]The total number of ways to form the committee is the product of these two combinations:
[binom{17}{6} times binom{15}{4} 12376 times 1365 16,893,240]Thus, there are 16,893,240 possible ways to form a committee of 10 members with exactly 6 seniors and 4 juniors from a pool of 17 seniors and 15 juniors.
Exploring the Calculation in Detail
Let's break down the calculation to see the step-by-step reasoning:
Step 1: Selecting Seniors
The number of ways to select 6 seniors from 17 can be calculated as follows:
(17 times 16 times 15 times 14 times 13 times 12 / 6 times 5 times 4 times 3 times 2 times 1 12376)
This simplifies to:
[binom{17}{6} 12376]Step 2: Selecting Juniors
The number of ways to select 4 juniors from 15 can be calculated as follows:
(15 times 14 times 13 times 12 / 4 times 3 times 2 times 1 1365)
This simplifies to:
[binom{15}{4} 1365]Step 3: Combining Both Selections
The total number of ways to form the committee is the product of the two combinations:
[binom{17}{6} times binom{15}{4} 12376 times 1365 16,893,240]Conclusion
In conclusion, the number of ways to form a committee of 10 members with exactly 6 seniors and 4 juniors from a pool of 17 seniors and 15 juniors is 16,893,240. This demonstrates the power of the combination formula in solving real-world problems involving selection and arrangement.