Problem 39 - Algebra and Trigonometry Real Mathematics, Real People

Question

Evaluate \(_{n} P_{r}\) using a graphing utility. $$_{120} P_{4}$$

Step-by-Step Solution

Answer

The solution to \(_{120}P_{4}\) is the number displayed on the graphing utility after performing the permutation calculation.

Step 1: Find the Permutation Function or Factorial Button On The Calculator

Depending on the calculator, there will either be a button or function for permutations or factorials. In case there's a permutation function (usually marked as nPr), inputs are 120 for 'n' and 4 for 'r'. If only a factorial function is available (normally denoted by n!), inputs are 120 for n and (120-4) or 116 for (n-r).

Step 2: Calculate The Permutation

If a permutation function is being used, simply input the values of 120 for n and 4 for r into the function: \( _{120}P_{4} \). If a factorial function is being used, calculate the factorials of n and (n-r), then divide the two. This should result into: \( \frac{120!}{116!} \)

Step 3: Note down the Result

Write down the result displayed on the calculator. This is the answer, which denotes how many different permutations of 4 items can be chosen from 120.

Key Concepts

Factorials

In mathematics, the concept of factorials is a foundational building block for permutations and combinations. A factorial, denoted by an exclamation mark (!), is the product of all positive integers up to a specified number. For example, the factorial of a number 'n', written as \( n! \), is calculated as:

• \( 5! = 5 \times 4 \times 3 \times 2 \times 1 = 120 \)

Factorials play a crucial role in determining permutations, as they provide a way to calculate the number of ways to arrange a set of items. Whenever you see a factorial, think of it as a way to list out all the possible orders of a set of elements.

In the context of the exercise, we used factorials to compute permutations using the formula \( \frac{n!}{(n-r)!} \). This is particularly important when a calculator does not have a direct permutation function, allowing the application of factorials to achieve the same result.

Graphing Utility

A graphing utility is a powerful calculator tool designed to solve mathematical equations, including those involving algebra, calculus, and statistics. Many modern graphing calculators come with built-in functions that simplify calculating permutations and combinations.

For the purpose of evaluating \(_{120} P_{4}\), a graphing utility can either directly use the permutation function, often indicated by the symbol 'nPr', or it can use the factorial function to compute permutations.

• If the calculator has an 'nPr' function, you can compute \( _{120}P_{4} \) directly by entering 120 as 'n' and 4 as 'r'.
• Alternatively, if the calculator only supports factorials, you can compute it using the formula \( \frac{120!}{116!} \).

Graphing utilities can save time and reduce errors in computation, making them an essential tool for students and professionals alike.

nPr

The 'nPr' function is a key feature in understanding permutations. In combinatorics, permutations refer to the arrangement of a set of objects in a specific order. The notation 'nPr' is used to determine how many ways we can select 'r' objects from 'n' total objects.

• The formula is expressed as \( nPr = \frac{n!}{(n-r)!} \).
• It indicates the importance of the order of selection, meaning \( _{120}P_{4} \) is not the same as \( _{4}P_{120} \).

In the exercise example, \( _{120}P_{4} \) calculates the number of ways to arrange 4 items out of 120. This underscores the flexibility and utility of the 'nPr' function in evaluating various scenarios in combinatorics.

Using a calculator with 'nPr' directly computes these scenarios efficiently, allowing a straightforward approach to permutation problems.

Combinatorics

Combinatorics is a branch of mathematics dealing with combinations, arrangements, and counting, especially related to constructing sets in a specific order. It covers several key principles including permutations, combinations, and factorials, among others.

In practical terms, combinatorics helps solve problems involving selecting items or arranging objects. For instance:

• Permutations are used when the order is significant, such as arranging books on a shelf.
• Combinations are for situations where the order does not matter, like selecting a team of players.

Understanding these concepts is vital for tackling a range of real-world problems, from optimizing routes to arranging seating plans. In this exercise, we focused on permutations because \( _{120}P_{4} \) required calculating the different ways four items can be chosen from 120 objects.

Whether you're using a graphing utility or manual calculations, grasping the principles of combinatorics opens up a wide range of problem-solving strategies.