Question

Evaluate the expression. \({ }_9 P_1\)

Short answer

The evaluation of the expression \({ }_9 P_1\) is 9.

Step-by-step solution

Understanding the Problem

The task here involves evaluating the permutation \({ }_9 P_1\). This means we are to find the number of ways of arranging 1 item from a total of 9 items.

Applying the Permutation Formula

We can apply the permutation formula \(P^n_r = \frac{n!}{(n-r)!}\), where n is the total number of items available and r is the number of items we are selecting. In our case, \(n = 9\) and \(r = 1\) which gives us \({ }_9 P_1 = \frac{9!}{(9-1)!}\)

Evaluating Factorials

A factorial of a number n, written as n!, is the product of all positive integers less than or equal to n. Hence, \(9! = 9 \times 8 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1\) and \((9-1)! = 8! = 8 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1\). We can then simplify the expression \({ }_9 P_1 = \frac{9!}{8!} = \frac{9 \times 8 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1}{8 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1}\)

Simplifying the Equation

By cancelling out similar terms in the numerator and the denominator, we get the result as \({ }_9 P_1 = \frac{9!}{8!} = 9\). Hence, there are exactly 9 ways of arranging 1 item from a total of 9 items.

Key concepts

Factorials

When faced with the task of finding the number of different ways we can arrange a certain number of items, factorials often come into play. But what exactly is a factorial? A factorial, represented by an exclamation point (!), is the product of all positive integers from 1 up to a given number. For example, the factorial of 5, written as 5!, would be computed as:
\(5! = 5 \times 4 \times 3 \times 2 \times 1 = 120\).
This operation is crucial when trying to understand permutations because it shows the total number of possible arrangements if we were to use all the available items. The understanding of factorials is foundational, and it allows us to evaluate permutation expressions effectively. In simpler cases, where we calculate the factorial of a number just above 1, like \(9!\) over \(8!\), a shortcut we can take is to realize that every term except the last one on the numerator and denominator cancels out. This leaves us with \(9!\) over \(8!\) equal to just 9, as seen in the example provided.

Arranging Items

The concept of arranging items is inherently related to permutations and plays a vital role in understanding them. Essentially, whenever we talk about arranging items, we refer to the number of different sequences that can be created by reordering a set of distinct items. Let's consider an example: if we have three books (A, B, C), they can be arranged in six different ways (ABC, ACB, BAC, BCA, CAB, CBA).
What's interesting is that as the number of items increases, the total number of arrangements grows rapidly—showcasing the power of permutations. Here's a useful tip: when dealing with problems requiring the arrangement of all items, the solution is simply the factorial of the number of items. However, when arranging a subset, the permutation formula is needed to find the correct number of arrangements.

Permutation Formula

To grasp permutations further, it's essential to understand the permutation formula. This formula, given by \(P^n_r = \frac{n!}{(n-r)!}\), helps us calculate the number of possible arrangements (permutations) of r items taken from a larger set of n distinct items. In the permutation formula, \(n\) stands for the total number of items, and \(r\) represents the number of items to arrange.
For instance, if we have a set of nine different books and wish to find out how many ways we can arrange three of them, we would use the permutation formula: \({ }_9 P_3 = \frac{9!}{(9-3)!} = \frac{9!}{6!}\).
To evaluate expressions involving permutations, it is crucial to understand factorial simplification, as it allows you to cancel out terms and arrive at the solution more quickly. In practice and for SEO optimization, remember that understanding the permutation formula empowers students to solve problems related to arranging items in a specific order, which is a common challenge in many math courses.