Problem 23 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

In a state lottery, there are 15 finalists eligible for the Big Money Draw. In how many ways can the first, second, and third prizes be awarded if no ticket holder can win more than one prize?

Step-by-Step Solution

Answer

There are 2730 ways to award the first, second, and third prizes among the 15 finalists if no ticket holder can win more than one prize. This is calculated using the permutation formula: P(15, 3) = 15! / 12! = 15 * 14 * 13 = 2730.

Step 1: Identify the problem type

This is a permutation problem because we care about the order of the prizes (first, second, and third).

Step 2: Use the Permutation Formula

The formula for permutations is: P(n, r) = n! / (n - r)! Where n is the number of elements, r is the number of elements we are choosing, and ! denotes a factorial.

Step 3: Plug in the values into the Permutation Formula

We need to find the permutations of 15 finalists (n) taken 3 at a time (r). P(15, 3) = 15! / (15 - 3)!

Step 4: Simplify the factorial expression

Now, let's simplify the expression. P(15, 3) = 15! / 12!

Step 5: Compute the answer

To compute the answer, we can cancel out the common 12! factor in both the numerator and the denominator: P(15, 3) = (15 * 14 * 13 * 12!) / 12! P(15, 3) = 15 * 14 * 13 Now, let's multiply these numbers to get the final answer: 15 * 14 * 13 = 2730 So, there are 2730 ways to award the first, second, and third prizes among the 15 finalists if no ticket holder can win more than one prize.

Key Concepts

Factorial

The concept of a factorial is fundamental in mathematics, especially when discussing permutations and combinations. The factorial of a non-negative integer, represented as \( n! \), is the product of all positive integers less than or equal to \( n \). For example, \( 5! = 5 \times 4 \times 3 \times 2 \times 1 = 120 \). Factorials grow very quickly with increasing numbers. The mathematical simplicity of factorial calculations is pivotal in many problems involving arrangements and permutations.Factorials are also involved in calculating permutations, helping to handle the "order of arrangement" without explicitly writing out all possibilities. In permutation problems like the one above, calculating something like \( 15! \) involves multiplying all integers from 1 to 15, but often large parts of the top and bottom of fractions cancel each other out, simplifying the calculation.

Permutation Formula

Permutations are all about arranging items where the order matters. That's why they are crucial in scenarios like awarding prizes to finalists. The permutation formula \( P(n, r) = \frac{n!}{(n-r)!} \) helps us calculate the number of ways to arrange \( r \) objects from a set of \( n \) objects.In the permutation formula:

• \( n \) is the total number of items you have.
• \( r \) is the number of spots you must fill or the exact number of items you need to arrange.

Using the permutation formula involves computing the factorial of both \( n \) and \( n-r \). In our lottery example, to find the various ways to award 3 prizes out of 15 finalists, you plug \( n = 15 \) and \( r = 3 \) into the formula, simplifying as necessary. The formula streamlines complex counting processes, ensuring accurate results without extensive listing.

Combinatorics

Combinatorics is the field of mathematics focused on counting, specifically regarding how items can be selected and arranged. It's a key area when considering permutations, as it deals with the "arrangement" and "selection" of items. Combinatorics principles are applied to understand and solve problems related to selections and orderings, such as lottery draws or award distributions. When the order of selection is important, as in our exercise, permutations are used. When the sequence doesn't matter, we use combinations instead, where the focus shifts from "who comes first" to "who is chosen." Combinatorics embraces these differences, providing tools to count possibilities in both contexts effectively. Understanding combinatorics, including both permutation and combination methods, empowers students to efficiently tackle various math problems involving selection and arrangement of items. This makes combinatorics an essential building block in probability and many real-world applications.