Problem 36 - College Algebra

Question

Use the formula for \(_{n} P_{r}\) to solve Exercises \(33-40\) Suppose you are asked to list, in order of preference, the three best movies you have seen this year. If you saw 20 movies during the year, in how many ways can the three best be chosen and ranked?

Step-by-Step Solution

Answer

The three best movies can be chosen and ranked in 6840 ways.

Step 1: Identify Variables

Identify the variables for the formula, where \( n \) is the total number of movies seen and \( r \) is the number of movies to rank. Here, n = 20 and r = 3.

Step 2: Calculate Factorials

Calculate the factorials for \( n \) and \( n-r \). Factorial is the product of an integer and all the integers below it. So, \( 20! = 20 \times 19 \times 18 \times \ldots \times 1 \) and \( (20 - 3)! = 17! = 17 \times 16 \times 15 \times \ldots \times 1 \).

Step 3: Apply the Permutation Formula

Now apply the permutation formula which is _nP_r = \( n! / (n - r)! \). So, _20P_3 = \( 20! / (20 - 3)! \) = \( 20! / 17! \)

Step 4: Simplify the Calculation

The values that are greater than 17 in 20! cancel out with 17!, leaving \( 20 \times 19 \times 18 \). So, _20P_3 = \( 20 \times 19 \times 18 \)

Step 5: Compute the Result

Finally, Multiply the remaining values to get the number of ways the three best movies can be chosen and ranked. _20P_3 = \( 20 \times 19 \times 18 = 6840 \)

Key Concepts

Factorial Notation

When we speak of factorial notation in algebra, we refer to a mathematical operation that involves multiplying a series of descending natural numbers. Symbolized by an exclamation point (!), the factorial of a non-negative integer 'n' is the product of all positive integers less than or equal to 'n'. For instance, the factorial of 5, denoted as \(5!\), is calculated as:

\[5! = 5 \times 4 \times 3 \times 2 \times 1 = 120\]

Factorial notation plays a crucial role in fields such as combinatorics, probability, and algebra. It's key for calculating permutations and combinations where order matters or doesn't matter, respectively.

In the example provided, calculating \(20!\) allows us to determine all the ways 20 movies can be arranged, a foundational step in finding out how many ways we can rank 3 of them. The factorial notation simplifies what would otherwise be a complex problem into manageable steps.

Combinatorics

Combinatorics is a branch of mathematics focused on counting, arranging, and finding patterns in sets of elements. It includes the study of permutations, combinations, and other mathematical structures. While permutations deal with the arrangement of all members of a set into some sequence or order, combinations refer to the selection of items from a set regardless of the order. Understanding the principles of combinatorics is essential for solving problems that require counting possible configurations, such as the number of ways to select and rank movies.

This field is incredibly useful not just in mathematical theory but also in practical applications such as algorithm design, cryptography, and probability theory. The movie selection problem is an example of a combinatorial problem, where we calculate the possible arrangements of a subset (three best movies) out of a larger set (all twenty movies seen).

Permutation Formula

The permutation formula is instrumental in solving problems involving the arrangement of items where the order is important. The formula is represented as \(_n P_r = \frac{n!}{(n-r)!}\), where 'n' is the total number of items and 'r' is the number of items to be arranged.

In the context of our example, to find out in how many ways we can rank three preferred movies out of twenty can be determined using the permutation formula. By substituting 'n' with 20 and 'r' with 3, we calculate according to the formula:

\[_{20} P_{3} = \frac{20!}{(20-3)!}\]

This simplifies the process of calculating the permutations as we divide the factorial of the total number of movies by the factorial of the difference between the total number of movies and the number to be ranked. The permutation formula enables a structured and precise way to figure out possible arrangements, ensuring that complex counting tasks can be tackled systematically.