Problem 25 - A Graphical Approach to Precalculus with Limits

Question

Use a calculator to evaluate each expression. \({ }_{20} P_{5}\)

Step-by-Step Solution

Answer

The value of _{20}P_{5} is 1860480.

Step 1: Understanding the Permutation Notation

The notation _{20}P_{5} refers to a permutation, which means the number of ways to arrange 5 items from a set of 20 items. The formula for permutations is given by _{n}P_{r} = \frac{n!}{(n-r)!}, where \(n\) is the total number of items and \(r\) is the number of items to arrange.

Step 2: Substituting Values into the Formula

In this problem, \(n = 20\) and \(r = 5\). Substitute these values into the permutation formula to find the number of permutations: _{20}P_{5} = \frac{20!}{(20-5)!} which simplifies to \frac{20!}{15!}.

Step 3: Calculating Factorials

Calculate \(20!\) and \(15!\) using a calculator.- \(20!\) is the product of all whole numbers from 20 down to 1.- \(15!\) is the product of all whole numbers from 15 down to 1.

Step 4: Simplifying the Expression

Calculate \(\frac{20!}{15!}\) using a calculator. We can simplify this to only calculate 20 imes 19 imes 18 imes 17 imes 16, as 15! cancels out in the fraction:20! = 20 imes 19 imes 18 imes 17 imes 16 imes 15!.15! cancels out in the numerator and the denominator.

Step 5: Final Calculation

Calculate 20 imes 19 imes 18 imes 17 imes 16 using a calculator to find the final value for _{20}P_{5}. This will give you the total number of ways to arrange 5 items out of 20.

Key Concepts

Factorial

Factorials are a fundamental aspect of permutations and many other mathematical concepts. The factorial of a number, denoted with an exclamation mark "!", is the product of all positive integers up to that number. For instance, the factorial of 3, written as \(3!\), is equal to \(3 \times 2 \times 1 = 6\).

Factorials grow rapidly. Even with relatively small numbers, calculations can become quite large. For example, \(5!\) equals \(5 \times 4 \times 3 \times 2 \times 1 = 120\). When applied to permutations, these calculations help determine the arrangements possible. In permutations like \( _{20}P_{5} \), factorials play a key role. You calculate \(20!\) and \(15!\) to find how many ways you can arrange 5 items out of 20.

Calculator Use

Using a calculator is essential for quickly evaluating permutations, especially when dealing with large numbers. Calculators typically have a factorial function, often labeled as "!", which simplifies these massive multiplications.

Let's say you need to calculate \(20!\). Instead of multiplying 20 down to 1 manually, enter 20 and press the "factorial" button on your calculator. This saves a lot of time and reduces the chance for errors.

• Ensure your calculator is in the correct mode to perform factorial operations.
• Double-check you have entered the right numbers before hitting enter.
• Some calculators directly allow permutation calculations by selecting \(_{n}P_{r}\) and entering your values for \(n\) and \(r\).

Using technology wisely can help make finding permutations accessible and efficient.

Permutation Formula

The permutation formula is used to find the number of possible arrangements of a given set of items. It is given by \(_{n}P_{r} = \frac{n!}{(n-r)!}\), where \(n\) is the total number of items and \(r\) is the number of items to arrange. This formula becomes especially useful when you need to determine how to arrange some elements from a larger group.

For example, with \(_{20}P_{5}\), we substitute \(n = 20\) and \(r = 5\) into our formula. This simplifies to \(\frac{20!}{15!}\), as \(n-r = 20-5 = 15\).

To simplify this calculation:

• Recognize that \(15!\) in the denominator cancels with \(15!\) in the numerator.
• This leaves us with \(20 \times 19 \times 18 \times 17 \times 16\).
• Calculate the remaining products to find the number of permutations.

This approach makes it easier to handle large factorial computations, allowing you to understand and compute permutations more effectively.