Problem 18 - College Algebra

Question

In Exercises \(17-20,\) 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 and there are 117,600 different ways the prizes can be awarded because the order of people winning the prize matters.

Step 1: Identify the nature of the problem

In the given problem, three different prizes are being awarded. The award for the first prize cannot be given to the second or third person, and likewise, the award for the second prize cannot be given to the third person. So, the order in which the awards are presented is essential, making this a permutation problem.

Step 2: Find the number of ways

In a permutation problem like this, the number of ways the prizes can be distributed is the permutation of 50 people taken 3 at a time. In mathematical notation, this can be calculated using the formula: \( P = \frac{n!}{(n-r)!} \), where \( n \) is the total number of people (50) and \( r \) is the number of prizes (3). Plugging the values, \( P = \frac{50!}{(50-3)!} \).

Step 3: Simplify the result

Simplifying the factorial terms, we obtain \( P = \frac{50 \cdot 49 \cdot 48}{3 \cdot 2 \cdot 1} \), which computes to 117,600

Key Concepts

Permutation Problem

Permutations are fundamental to understanding how different sets of items can be arranged where order matters. In most cases involving permutations, each element in a set is unique and the sequence in which they are arranged is important. For instance, when giving out prizes in a raffle, as in the exercise provided, the first, second, and third prizes are distinct from one another, meaning the same prize cannot be awarded to more than one person. Consequently, the arrangement of the winners is a permutation problem because we are not just selecting winners; rather, we are arranging them in a specific order.

The factorial notation comes in handy for solving these kinds of problems, as it allows us to easily calculate the number of possible arrangements without having to list them all out. A good tip for solving permutation problems is to first establish whether the order of items or elements matters. If it does, then what you're dealing with is a permutation problem. Next, use factorial notation to express the number of ways to arrange the items, taking into account that the order is crucial.

Factorial Notation

Factorial notation is an essential mathematical concept, especially when dealing with permutations and combinations. Factorial of a non-negative integer, denoted with an exclamation mark (!), is the product of all positive integers less than or equal to that number. For example, the factorial of 5, written as '5!', is calculated as 5 × 4 × 3 × 2 × 1. This equals 120. In general, for any non-negative integer 'n', the factorial is given as:\[ n! = n \times (n - 1) \times (n - 2) \times \ \times 3 \times 2 \times 1. \]

A useful tip when working with large numbers is to remember that the factorial of a number grows extremely fast—so fast that even the factorial of 10 is already 3,628,800. This is why factorial notation is used in permutations and combinations, where we are often faced with a large set of items to arrange or select from. For instance, calculating the different ways 50 raffle tickets can be distributed involves substantially large numbers, and thus factorial notation is invaluable.

Raffle Probability

Raffle probability is a practical application of permutations and combinations in probability theory that deals with the likelihood of different outcomes in a raffle draw. When computing the raffle probability, especially when there are multiple prizes of different values like in the original exercise, it's critical to consider permutations because the prizes are ranked and the order in which they are awarded is significant. Remember that the basic principle of probability is outcomes over possibilities. To elaborate, in a scenario where we award three prizes from a pool of fifty tickets, we are looking for all the possible ways those prizes can be distributed among the participants.

As with the provided example, if first prize is distinct from second prize, which is also distinct from third prize, the number of different ways to award these prizes can be calculated using permutations. Keep in mind that the probability of any single arrangement occurring is one divided by the total number of permutations. Enhancing the understanding of raffle probability can greatly benefit students in grasping more complex probability problems and scenarios.