Problem 18 - College Algebra

Question

Does the problem involve permutations or combinations? Explain your answer (It is not necessary to solve the problem. Fifty people purchase raffle tickets. Three winning tickets are selected at random. If first prize is \(\$ 1000,\) second prize is \(\$ 500\) and third prize is \(\$ 100,\) in how many different ways can the prizes be awarded?

Step-by-Step Solution

Answer

The problem involves permutations as the order of the winners matters for the distribution of the first, second, and third prizes. The number of different ways the prizes can be awarded is calculated using the formula for permutations and results in \(\frac{50!}{(50-3)!}\) different ways.

Step 1: Identify the type of problem

Firstly, determine whether the problem involves permutations or combinations. In this case, the problem deals with permutations since order matters - the first, second, and third prizes are distinct. Thus, the problem involves permutations and not combinations.

Step 2: Use the formula for permutations

The formula for permutations when order matters is given by \(P(n,r) = \frac{n!}{(n-r)!}\). In this situation, \(n\) represents the number of raffle ticket purchasers (50) and \(r\) represents the number of prizes to be won (3).

Step 3: Apply the values to the formula

Substitute the values into the permutations formula. Hence, the number of ways the three prizes can be awarded is given by \(P(n,r) = \frac{50!}{(50-3)!}\).

Key Concepts

Factorial Notation

The concept of factorial notation is the cornerstone of calculating permutations in probability. It is represented by the exclamation mark (!) following a non-negative integer. Factorial notation essentially tells us to multiply a series of descending positive integers. For example, the factorial of 5, denoted as 5!, equals
\(5 \times 4 \times 3 \times 2 \times 1 = 120\).
The factorial of 0 is defined as 1. Factorials are fundamentally linked to permutations because they allow us to count the number of different ways we can arrange a set of distinct items without repetition. When we calculate permutations, we use factorials to consider all the possible arrangements for a given number of objects.

Permutations Formula

When we want to determine the number of ways of arranging or ordering a specific number of items from a larger set, we use the permutations formula. It's crucial when order is significant in the arrangement of the items. The formula for permutations, denoted as \(P(n,r)\), is
\[ P(n, r) = \frac{n!}{(n-r)!} \]
Where \(n\) represents the total number of items to choose from, and \(r\) is the number of items we want to arrange.

Here's a Practical Example:

Consider selecting a president, vice-president, and secretary from a board of 10 members. The order of selection is important (since the positions are different), and we can use the permutations formula to find the number of ways to pick and arrange these officers. In this case, we calculate it as \(P(10, 3)\).

Applying the formula, we get
\[ P(10, 3) = \frac{10!}{(10-3)!} = \frac{10!}{7!} \]
Simplifying this, we find the total number of different ways the officers can be arranged.

Ordering Events in Probability

Ordering events in probability is particularly important when dealing with sequences where the arrangement or sequence of outcomes matters. This concept is exemplified when considering the probability of sequential events such as rankings in a competition or the order in which balls are drawn from a container.

Relation to Permutations:

The permutations formula comes into play as it helps us to quantify the possible sequences of events. In a scenario where we have multiple events occurring one after the other, the permutations formula provides a systematic way to calculate the total number of possible orders.

For instance, if we have three races and want to know the number of possible ways to award first, second, and third place out of eight runners, permutations allow us to identify all the possible outcomes since the order of finishing is crucial. An in-depth understanding of ordering is significant for accurately interpreting probabilities in various real-world situations.