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

The awards can be given out in 2,730 different ways.

Step 1: Understanding the Problem

We need to determine the number of ways to distribute three distinct awards among 15 students. Here, the order in which the awards are given matters, and a student can only receive one award.

Step 1: Choose the First Awardee

To choose the recipient of the first award, we have 15 possible choices since any of the 15 students can receive the award.

Step 2: Choose the Second Awardee

After giving the first award, there are 14 students left who can receive the next award. Therefore, for the second award, we have 14 options.

Step 3: Choose the Third Awardee

After giving out the second award, there are now 13 students remaining. Thus, for the third award, there are 13 choices left.

Step 5: Final Calculation

Multiply the number of choices at each step: the total number of ways to distribute the awards is calculated as \(15 \times 14 \times 13\).

Key Concepts

Permutations

Permutations are arrangements of objects or items in a specific order. For example, if you have three distinct awards and 15 students, as in the exercise, you want to determine how many different ways you can distribute these awards.

In permutations, the order matters. This means that distributing award A, then award B, then award C to students X, Y, and Z will be considered different from distributing award B, then award A, then award C.

When dealing with permutations, it's crucial to decrement the number of choices available as each selection is made. If a student receives an award, they should not be considered for additional awards, retaining the idea that no repetition of awardees occurs. Thus, in our example, after the first award is given to one of 15 students, only 14 students remain eligible for the next award, and after that, only 13 remain eligible for the final award.

The formula for permutations of choosing "r" objects out of a set of "n" items is denoted by:

• \[P(n, r) = \frac{n!}{(n-r)!}\]

In the exercise, it's a case of finding the number of ways to distribute 3 distinct awards among 15 students, which fits into the permutation formula as:

• \[P(15, 3) = \frac{15!}{(15-3)!}\]

This simplifies to multiplying the top three numbers, as shown in the step-by-step solution: \(15 \times 14 \times 13\).

Combinatorial Counting

Combinatorial counting involves determining the number of different ways to arrange or select items from a set. It's a key concept in mathematics that applies to various real-world scenarios, including award distributions, seating arrangements, and DNA sequencing.

The principle of combinatorial counting simplifies complex problems by breaking them into smaller, manageable steps. By focusing on one decision at a time and considering constraints, like excluding previous selections, you can systematically count possibilities.

• One fundamental law used is the multiplication principle: If there are \(n_1\) ways to make the first choice, \(n_2\) ways to make the second choice, and \(n_k\) ways for the k-th choice, the total number of ways to make all choices is given by \(n_1 \times n_2 \times \cdots \times n_k\).
• Another helpful concept is understanding when to use permutations versus combinations. Combinations ignore order, but permutations require you to focus on every possible sequence.

In the context of the exercise, combinatorial counting is employed to determine how many ways three distinct awards can be assigned to students where different sequences of awardee receive different considerations.

Discrete Mathematics

Discrete mathematics deals with distinct and separate values, often counted in whole numbers rather than continuous spectra. It's the backbone of computer science, cryptography, and many areas of quantitative analysis.

A primary focus of discrete mathematics is structurally organizing objects and systems in a countable manner. Permutations and combinations are just some of the numerous discrete structures used to simplify complex problems.

In the context of the award distribution scenario, discrete mathematics offers principles such as permutations to systematically and methodically solve the counting problem. It's all about how we organize these awards among students without overlap, ensuring each student receives at most one award.

• This discipline often involves working with algorithms, theorems, and logical structures.
• Students are challenged to think critically about each step, focusing on discrete choices and adopting strategies to ensure no element is double-counted.

Breaking the problem into a series of independent decisions mirrors real-life scenarios, making discrete mathematics a practical and essential field of study.