Problem 15 - Calculus for Biology and Medicine

Question

Three different awards are to be given to a class of 15 students. Each student can receive at most one award. Count the number of ways these awards can be given out.

Step-by-Step Solution

Answer

There are 2730 ways to distribute the awards.

Step 1: Identify the Problem Type

We are dealing with a counting problem where we need to determine how many ways we can distribute three distinct awards to a group of students where each student gets at most one award. This is a permutation problem because the awards are distinct and order matters.

Step 2: Calculate the First Choice

For the first award, we can choose any one of the 15 students in the class to receive the award. This gives us 15 possible choices.

Step 3: Calculate the Second Choice

After the first student is given an award, 14 students remain without an award. Therefore, for the second award, we have 14 choices.

Step 4: Calculate the Third Choice

Once the first two awards have been given, 13 students remain without an award. Hence, we have 13 choices for the third award.

Step 5: Compute the Total Number of Permutations

To find the total number of ways to distribute the awards, multiply the number of choices for each award: \[15 \times 14 \times 13 = 2730\]

Key Concepts

Counting Problems

Counting problems are a fundamental aspect of mathematics, specifically when dealing with questions that require finding the number of ways something can be arranged or organized. In the context of awards distribution, counting problems help identify how different arrangements can occur given specific conditions.

• It's essential to determine whether order matters; if it does, we're dealing with permutations.
• Counting helps address problems involving limited quantities or conditions, such as each student receiving at most one award.
• Breaking the problem into smaller steps helps simplify complex counting tasks, as seen with awards.

Understanding counting problems can often involve recognizing patterns or utilizing mathematical tools, such as factorials or combinations, to calculate total possibilities.

Distinct Awards

When dealing with awards that are distinct, each award is unique and different from the others. This means that the order in which awards are given out matters significantly.

• For instance, giving Awards A, B, and C to students Jane, John, and Joe would be a different arrangement than giving A to John, B to Joe, and C to Jane.
• This distinct characteristic leads us to use permutation methods, as opposed to combinations, where order isn't important.
• Identifying awards as distinct helps guide the approach for calculating the number of possible distributions.

By understanding that the awards are distinct, we apply suitable mathematical techniques to solve the problem effectively, ensuring we capture all possible outcomes.

Combinatorics

Combinatorics is the branch of mathematics that deals with counting, arrangement, and combination within a set of elements. It is a powerful tool used in calculating possible arrangements where restrictions are involved.

• In our problem, combinatorics helps assess how to distribute three distinct awards among 15 students while respecting the restriction that one award goes to one student only.
• The concept of a permutation arises due to the distinct nature of these awards and the importance of order.
• Utilizing combinatorics provides a systematic approach to what might seem like a daunting problem by breaking it into steps (using permutations) for clarity.

Combinatorics not only simplifies such counting problems but also enhances understanding of mathematical logic, paving the way for reasoning through similar challenges.