Problem 20 - 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 each prize is \(\$ 500,\) in how many different ways can the prizes be awarded?

Step-by-Step Solution

Answer

The problem involves combinations, not permutations. The number of different ways the prizes can be awarded is calculated using the combination formula \( C(50, 3) = \dfrac{50!}{3!(50-3)!} \).

Step 1: Determine if the problem is about permutations or combinations.

Because the problem doesn't specify an order in which the tickets will be selected or awarded, it suggests the process of combination. So, this problem involves combinations.

Step 2: Identify the total number of items and the number to be chosen.

The total number of raffle tickets, or our possible number of outcomes, is 50. The number of tickets to be chosen, or our outcomes of interest, is 3.

Step 3: Calculate the number of ways the prizes can be awarded.

The number of ways to select three winners from fifty people, which represents the number of different ways the prizes can be awarded, is calculated by the combination formula \( C(n, r) = \dfrac{n!}{r!(n-r)!} \), where \( n \) is the total number of items, \( r \) is the number of items to choose, and \( ! \) denotes factorials. Substituting our numbers into the formula gives \( C(50, 3) = \dfrac{50!}{3!(50-3)!} \).

Key Concepts

Permutations

In mathematics, permutations refer to the arrangement of objects in a specific order. The concept of permutations becomes important when the sequence in which items are arranged matters. For example, consider arranging the letters A, B, and C: the sequences ABC and CAB would count as two different permutations because the order is different. Permutations are calculated using the formula for permutations, which is given by: \[ P(n, r) = \frac{n!}{(n-r)!} \]Here, \( n \) is the total number of items, and \( r \) is the number of items to be arranged. Factorials are directly involved here, with \( n! \) being the product of all positive integers up to \( n \). Permutations are mainly used for scenarios where order is of significance such as seating arrangements, race ranks, or any sequential line-up situations.

Factorials

Factorials are a key component in combinatorics, significantly influencing calculations of permutations and combinations. The factorial of a non-negative integer \( n \), written as \( n! \), is the product of all positive integers less than or equal to \( n \). For example:- \( 5! = 5 \times 4 \times 3 \times 2 \times 1 = 120 \)- \( 3! = 3 \times 2 \times 1 = 6 \)And by definition, \( 0! = 1 \).Factorials rapidly increase in size as \( n \) grows, which is particularly important in combinatorics as they help denote the number of ways to arrange a set of objects.Understanding the growth and behavior of factorials helps one grasp how permutations and combinations scale with larger data sets.

Combinatorial Mathematics

Combinatorial mathematics mainly deals with counting or arranging combinations and permutations. It forms the backbone of probability and helps in solving problems related to arrangement and selection.
At the heart of combinatorial mathematics is the concept of combinations, especially relevant when the order does not matter. In our example exercise, selecting three winning raffle tickets from a group of fifty, without concern for order, is a classic combination problem.Combinations are calculated using:\[ C(n, r) = \frac{n!}{r!(n-r)!} \]This formula shows the number of ways to choose \( r \) items from \( n \) items without regard to order.
Combinatorial mathematics often requires both understanding the specific question of order importance, and applying appropriate formulas to calculate resultant possibilities effectively.