Problem 26 - A Graphical Approach to Precalculus with Limits

Question

Use a calculator to evaluate each expression. \({ }_{100} P_{5}\)

Step-by-Step Solution

Answer

\({ }_{100} P_{5} = 9,034,502,400\)

Step 1: Understand the Notation

The notation \({ }_{n} P_{r}\) refers to the number of permutations of \(n\) items taken \(r\) at a time. A permutation is an arrangement of objects in a specific order. This can be calculated using the formula: \({ }_{n} P_{r} = \frac{n!}{(n-r)!}\). In this exercise, \(n = 100\) and \(r = 5\).

Step 2: Apply the Permutation Formula

Substitute \(n = 100\) and \(r = 5\) into the permutation formula: \[{ }_{100} P_{5} = \frac{100!}{(100-5)!} = \frac{100!}{95!}\].

Step 3: Simplify the Expression

The expression \(\frac{100!}{95!}\) implies that we only need to compute the product of the first 5 terms of 100 factorial: \(100 \times 99 \times 98 \times 97 \times 96\). This simplifies the calculation by eliminating the repetition of 95! in the numerator and denominator.

Step 4: Calculate the Product

Use a calculator to multiply the values: \(100 \times 99 \times 98 \times 97 \times 96 = 9034502400\). This is the total number of permutations of 100 items taken 5 at a time.

Key Concepts

Factorial

Factorial is a fundamental concept in mathematics, particularly in permutations and combinatorics. It is denoted by an exclamation mark next to a number, like this: \(n!\). The factorial of a number \(n\) is the product of all positive integers from 1 up to \(n\). For example, \(5! = 5 \times 4 \times 3 \times 2 \times 1 = 120\).
Factorials are incredibly useful because they represent the number of different ways to arrange \(n\) distinct objects into a sequence.
Some specific points about factorials:

• \(0!\) is defined to be 1. This might seem counterintuitive, but it is useful in simplifying formulas in combinatorics.
• The factorial function grows very rapidly. For instance, while \(5! = 120\), \(10! = 3,628,800\), which is already a much larger number.
• Calculating large factorials manually is difficult, but calculators or software can handle these computations quickly.

Understanding factorials is crucial for computing permutations and combinations effectively, as they form the core part of the formulas used.

Combinatorics

Combinatorics is an area of mathematics focused on counting, arrangement, and combination of sets. It deals with questions of counting the number of ways certain patterns can be formed. This field is foundational for understanding permutations, combinations, and related structures.

There are two main categories in combinatorics:

• Combinations: When the order of selection doesn't matter. For instance, if you are choosing 3 fruits from a basket of 5 different fruits, the array of fruits chosen is important but not their order.
• Permutations: When the order does matter. This is the primary focus when dealing with problems like our exercise, which involves arrangements in specific orders.

Combinatorics are not only useful in mathematics but also in fields like computer science, statistics, and beyond. By understanding combinatorics, you gain a toolkit for addressing various real-world counting problems efficiently.

Permutation formula

In permutations, we're concerned with the arrangement of objects in a specific sequence, and this is where the permutation formula comes into play. The formula \( { }_{n} P_{r} = \frac{n!}{(n-r)!} \) allows us to determine the number of permutations of \(n\) items taken \(r\) at a time. This formula is derived from the concept of factorial.

Here's how it works:

• \(n!\), or "n factorial," calculates the total number of permutations of \(n\) items.
• The subtraction \((n-r)!\) accounts for the arrangements of the remaining \(n-r\) items not included in the permutation.
• Thus, by dividing the total number of permutations of \(n\) items by the permutations of the \(n-r\) items, we only count the arrangements of interest, which are those involving \(r\) selected items.

By applying this formula, one can efficiently compute large permutation problems, like the arrangement of 100 items taken 5 at a time, which becomes much easier with the factorial structure simplifying the calculation.