Problem 20 - Algebra and Trigonometry Real Mathematics, Real People

Question

Use a graphing utility to find \(_{n} C_{r^{*}}\) \(_{500}{C}_{498}\)

Step-by-Step Solution

Answer

The output of \(_{500}{C}_{498}\) should read as 124750. This is because in a group of 500, there are 124750 unique ways to choose 498 items.

Step 1: Setting up the calculator

First, you want to make sure you've got your graphing calculator at hand. To calculate a combination, you should find the function usually denoted as nCr, nPr or C(n,r).

Step 2: Input the combination

After identifying the correct function, input the numbers into the calculator in the appropriate order. In this case, our larger number is \( n = 500 \) and our smaller number is \( r = 498 \) . So, we input \(_{500}{C}_{498}\) into the calculator.

Step 3: Execute calculation

Finally, we execute the calculation by pressing the 'equals' button or other command that would give you the result of the calculation.

Key Concepts

Graphing Calculator

Graphing calculators are powerful tools that are especially beneficial in solving a variety of mathematical problems. When dealing with large numbers or complex calculations, such as combinations or permutations, these calculators simplify the task significantly. A graphing calculator doesn't just plot graphs, it also houses advanced mathematical functions, including statistical calculations, algebra, and calculus functionalities.

To use a graphing calculator for combination calculations, you first need to familiarize yourself with its function keys. Most graphing calculators have a dedicated math menu where special functions like nCr (for combinations) or nPr (for permutations) can be found. Once located, these can be accessed through special button combinations or through a catalog of functions. Graphing calculators save time and reduce human error when calculating combinations, making them an essential tool for students tackling probability and combinatorial problems.

Combinatorics

Combinatorics is the branch of mathematics dealing with combinations and permutations. It is crucial in fields where probability and decision-making are involved, such as statistics, computer science, and game theory. Simply put, combinatorics helps us understand how to count or list items in a structured manner.

Combinations are a fundamental concept in combinatorics. Unlike permutations, where order matters, combinations focus only on the selection of items, with no regard to their arrangement. This is why learning combinatorial concepts, like combinations, is vital. Knowing how many different ways you can choose items from a larger group is a skill that aids in probabilistic analyses. To calculate combinations, the formula used is:

• \(_{n}C_{r} = \frac{n!}{r!(n - r)!}\)

This formula means you're calculating how many ways you can select \(r\) items from a total of \(n\), without considering the order of the selection.

nCr Function

The nCr function is a special mathematical function used for calculating combinations. It is often seen as \(nCk\) or \(_{n}C_{r}\) in mathematical notation, and is pronounced "n choose r." This function determines how many ways you can choose \(r\) items from a set of \(n\) items. Let's break down its components:

• **n** is the total number of available items.
• **r** is the number of items to be chosen.

To solve \(_{500}{C}_{498}\), you enter the larger number first, followed by the smaller. This is important because the formula involves dividing factorials, which represent the product of an integer and all the integers below it. The nCr function in calculators abstracts this computation, allowing for quick and error-free results.

Using the nCr function through a calculator simplifies solving complicated problems in homework or exams, ensuring you spend less time on manual calculations and more on understanding the underlying math principles. As with our example, entering \(_{500}{C}_{498}\), and executing the calculation will give you the number of combinations possible, faster than manual computation.