Problem 62 - College Algebra

Question

Solve by the method of your choice. Fifty people purchase raffle tickets. Three winning tickets are selected at random. If each prize is \(\$ 500,\) in how many different ways can the prizes be awarded?

Step-by-Step Solution

Answer

Thus, the prizes can be awarded in 19,600 different ways.

Step 1: Understand the problem

This problem requires us to find the number of different ways the prizes can be awarded to three different people who bought tickets. The order does not matter because the three prizes are identical.

Step 2: Apply the combinations formula

The formula for combinations without repetition is \( C(n, r) = n! / [ r!(n-r)! ] \) where \(n\) represents the total number of options, \(r\) is the number of options to be selected, and \( ! \) denotes the factorial operation. Here, \(n\) = 50 and \(r\) = 3.

Step 3: Calculate the factorial for \(n\), \(r\), and \(n-r\).

Factorial of \(n\): \(n!\) = 50! ; Factorial of \(r\): \(r!\) = 3! ; Factorial of \(n-r\): \((n-r)!\) = 47!.

Step 4: Compute the total number of combinations

Substitute the values into the formula and calculate: \( C(50, 3) = 50! / [3! * 47!] \). This simplifies to \( (50*49*48) / (3*2*1) \) which equals to 19,600.

Key Concepts

Combinatorics

Combinatorics is a field of mathematics primarily concerning the study of finite or countable discrete structures. It involves counting, arranging, and analyzing all the possible configurations of a set of items according to specified rules. In our daily lives, combinatorics manifests in a variety of scenarios, such as determining possible combinations of pizza toppings or figuring out different ways to organize books on a shelf.

In formal education, combinatorics problems typically revolve around determining the number of possible groupings (combinations) or arrangements (permutations) of a certain number of items. For instance, when deciding how many different ways you can award prizes in a raffle or how many unique password combinations can be created from a given set of characters. Understanding the basic principles of combinatorics is essential in fields as diverse as computer science, statistics, and even biology.

Factorial Operation

The factorial operation is denoted by an exclamation point (!) and signifies the product of an integer and all the integers below it down to one. For instance, the factorial of 5, written as 5!, would be calculated as 5 x 4 x 3 x 2 x 1, which equals 120. Factorials are crucial in combinatorics because they provide a way to calculate permutations and combinations. Factorials grow extremely fast with each additional number, and their properties are pivotal in simplifying complex combinatorial expressions.

Understandably, computing factorials for large numbers can be arduous and tiresome. Thus, when it comes to permutations or combinations involving large numbers, there are ways to cancel out parts of the expression to make calculations more manageable, which is illustrated in the textbook problem's solution.

Permutations and Combinations

Permutations and combinations are two fundamental concepts in combinatorics that deal with the arrangement and selection of objects. When the order of selection is important, we use permutations. For example, the arrangement of books on a shelf where their sequence matters. Conversely, combinations refer to when the order of selection does not matter, such as picking a team of players or winners in a raffle, where the concern is who is on the team, not the order in which they were chosen.

In the given exercise, the combinations formula is utilized because the order of prize winners is not relevant; any group of three winners is as good as another. Improving comprehension of these concepts requires understanding that the formula for combinations, \( C(n, r) = \frac{n!}{r!(n-r)!} \) where \( n \) represents the total number of options and \( r \) is the number of options to be selected, is designed to account for the fact that order doesn't matter. This fundamental difference between permutations and combinations often clarifies for students when to use each concept in problem-solving.