When faced with a combinatorial problem like this, it’s essential to understand the underlying principles of permutations and combinations. This problem specifically deals with permutations, as the order in which people are chosen for the positions matters. We are selecting and arranging individuals for distinct roles, making the sequence of selection significant. Let's delve into the problem and its solution in detail. We aim to explain the concept clearly and make it easily understandable.
Understanding the Problem
At its core, this is a permutation problem. Permutations deal with the arrangement of items or people in a specific order. Here, we have 15 different people, and we need to fill three distinct positions. The keyword here is “different positions,” which implies that the order in which we select people matters. If we chose John for the first position, Mary for the second, and Peter for the third, it’s a different arrangement than choosing Mary for the first position, Peter for the second, and John for the third. Therefore, we need to calculate the number of permutations, not combinations.
Permutation vs. Combination
It’s crucial to differentiate between permutations and combinations. In combinations, the order of selection does not matter. For example, if we were simply choosing a committee of three people out of 15, without assigning specific roles, it would be a combination problem. However, since our problem involves distinct positions, we use permutations.
The Positions
We have three positions to fill. Let’s call them Position 1, Position 2, and Position 3. Each position needs to be filled by a different person from the pool of 15 individuals. The selection process happens sequentially. For the first position, we have a choice from all 15 people. Once the first position is filled, we move on to the second, where our pool of candidates has decreased. Then we fill the last position with the remaining pool of candidates.
Solving the Problem Step by Step
Let’s break down the selection process step by step to understand how we arrive at the solution. This step-by-step approach will help clarify the reasoning behind the correct answer and why other options are incorrect.
Step 1: Filling the First Position
For the first position (let's say, the President), we can choose any one of the 15 people. So, we have 15 options for the first position. This is our starting point. The key here is that each person is unique, and each can potentially fill this first role. The selection of the first person reduces the pool of candidates for subsequent positions.
Step 2: Filling the Second Position
Once we’ve filled the first position, we are left with 14 people. This is because one person has already been selected and cannot be chosen again for another position (since the same person cannot hold two different positions simultaneously). Thus, for the second position (perhaps the Vice President), we have 14 choices. The pool of candidates has shrunk, making each subsequent choice dependent on the previous one.
Step 3: Filling the Third Position
After filling the first two positions, we have used up two people from the original 15. This leaves us with 13 people to choose from for the third position (maybe the Treasurer). So, we have 13 options for the third position. The sequential nature of this selection process is crucial in understanding why we multiply these numbers together.
The Multiplication Principle
The fundamental principle of counting, often called the multiplication principle, is applied here. It states that if there are ‘m’ ways to do one thing and ‘n’ ways to do another, then there are m × n ways to do both. This principle extends to multiple events. In our case:
- There are 15 ways to fill the first position.
- For each of those 15 ways, there are 14 ways to fill the second position.
- For each of the combinations of the first two positions, there are 13 ways to fill the third position.
Therefore, the total number of ways to fill all three positions is the product of these choices: 15 × 14 × 13.
Calculating the Answer
Now that we’ve established the correct approach, let’s calculate the final answer. We multiply the number of choices for each position:
15 × 14 × 13 = 2730
So, there are 2730 different ways to fill the three positions by choosing from 15 different people.
Why Other Options Are Incorrect
It’s important to understand why the other options provided are not correct. This will reinforce the understanding of permutations and help avoid similar mistakes in the future.
- A) 15 * 3: This option simply multiplies the number of people by the number of positions. It doesn’t account for the decreasing number of choices as positions are filled. It treats each position independently, which is incorrect.
- B) 3 * 2 * 1: This option calculates the number of ways to arrange three items (or people) among themselves. It would be relevant if we had already selected three people and wanted to know how many ways they could be arranged in the positions, but it doesn’t address the initial selection from 15 people.
- D) 15 * 14 * 12 * 3: This option incorrectly includes the number 12 and multiplies by 3, which doesn’t logically fit the problem. The sequence should be a decreasing series of choices (15, 14, 13), not an arbitrary set of numbers.
Generalizing the Permutation Formula
This problem is a specific case of permutations. In general, the number of permutations of ‘n’ items taken ‘r’ at a time is denoted as P(n, r) and is calculated using the formula:
P(n, r) = n! / (n - r)!
Where:
- n! (n factorial) is the product of all positive integers up to n.
- r is the number of items being chosen and arranged.
In our case, n = 15 (the total number of people) and r = 3 (the number of positions). Applying the formula:
P(15, 3) = 15! / (15 - 3)! P(15, 3) = 15! / 12! P(15, 3) = (15 × 14 × 13 × 12!) / 12! P(15, 3) = 15 × 14 × 13
This confirms our earlier calculation, showing how the permutation formula provides a generalized method for solving such problems. Understanding this formula is valuable for tackling a variety of permutation questions.
Real-World Applications
Permutation problems are not just theoretical exercises; they have practical applications in various fields. Recognizing these real-world scenarios helps to appreciate the relevance of combinatorial mathematics. Here are a few examples:
Scheduling
Consider scheduling tasks in a specific order. If you have a set of tasks and need to determine the number of ways to sequence them, you are dealing with a permutation problem. The order in which tasks are performed can have significant implications for efficiency and outcomes.
Password Generation
In computer science, permutations are used in generating passwords and encryption keys. The security of a password depends on the number of possible combinations, which is a permutation problem when the order of characters matters.
Team Formation with Roles
Similar to our initial problem, forming a team where each member has a distinct role (e.g., captain, vice-captain, etc.) involves permutations. The order of selection determines the role each person will play.
Arranging Items
Consider arranging books on a shelf or songs in a playlist. The number of different arrangements is a permutation problem. Each unique arrangement can have a different aesthetic or functional value.
Tips for Solving Permutation Problems
To effectively solve permutation problems, consider these tips:
Identify if Order Matters
The most crucial step is to determine whether the order of selection matters. If it does, you’re dealing with a permutation; if not, it’s a combination.
Break Down the Problem
Divide the problem into sequential steps. For each step, determine the number of choices available. This makes the problem easier to visualize and solve.
Use the Multiplication Principle
Apply the multiplication principle to find the total number of ways by multiplying the number of choices at each step.
Apply the Permutation Formula
For more complex problems, use the permutation formula P(n, r) = n! / (n - r)! to calculate the answer directly.
Practice, Practice, Practice
The more permutation problems you solve, the better you’ll become at recognizing patterns and applying the correct methods. Use textbooks, online resources, and practice questions to hone your skills.
Conclusion
The problem of filling three different positions from 15 people is a classic example of a permutation problem. The key to solving it lies in understanding that the order of selection matters and applying the multiplication principle. By breaking the problem down into steps and recognizing the sequential nature of the choices, we can accurately calculate the number of ways to fill the positions. The correct answer, 15 × 14 × 13, or 2730, highlights the importance of permutations in combinatorial mathematics and its wide range of applications in real-world scenarios. By understanding the underlying principles and practicing problem-solving techniques, you can confidently tackle similar questions and appreciate the power of combinatorial thinking.
The correct answer is C) 15 * 14 * 13.