Problem 16 - A Graphical Approach to College Algebra

Question

Evaluate each expression. $$P(10,3)$$

Step-by-Step Solution

Answer

The value of \(P(10,3)\) is 720.

Step 1: Understanding Permutations

The expression \(P(10,3)\) represents the number of permutations of 10 items taken 3 at a time. A permutation is an ordered arrangement of items, where the order matters.

Step 2: Applying the Permutation Formula

The formula for a permutation \(P(n, r)\), where \(n\) is the total number of items and \(r\) is the number of items to be arranged, is given by: \[ P(n, r) = \frac{n!}{(n-r)!} \] In this exercise, \(n = 10\) and \(r = 3\).

Step 3: Calculating Factorials

Calculate the factorials: For \(n = 10\): \[ 10! = 10 \times 9 \times 8 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1 \]For \(n-r = 7\): \[ 7! = 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1 \]

Step 4: Substitute and Simplify

Substitute the factorials into the permutation formula: \[ P(10, 3) = \frac{10!}{7!} = \frac{10 \times 9 \times 8 \times 7!}{7!}\] Cancel out \(7!\) from the numerator and the denominator:\[ P(10, 3) = 10 \times 9 \times 8 \]

Step 5: Final Calculation

Perform the multiplication:\[ 10 \times 9 = 90 \]\[ 90 \times 8 = 720 \]So, \(P(10, 3) = 720\).

Key Concepts

Factorials

Factorials are a fundamental concept in mathematics, particularly useful in permutations and combinations. A factorial is denoted by the symbol "!" and represents the product of all positive integers up to a given number. For example, the factorial of 5, written as \(5!\), is calculated as \(5 \times 4 \times 3 \times 2 \times 1\), which equals 120.
Factorials grow quickly as the numbers increase, as each factorial is the previous number’s factorial multiplied by a new integer. For instance:

• \(4! = 24\)
• \(5! = 120\)
• \(6! = 720\)

Understanding how factorials work is key when diving into permutations, as they play a crucial role in calculating the number of possible arrangements of a set of items. In permutation calculations like \(P(10, 3)\), factorials help determine how many different ways you can order a selection from a larger group.

Permutation Formula

Permutations refer to the arrangement of items in a specific order. When working on permutation problems, the order in which you arrange these items matters. For example, the sequence ABC is different from BCA.
The permutation formula is given by:\[P(n, r) = \frac{n!}{(n-r)!}\]where \(n\) is the total number of items you have to pick from, and \(r\) is the number of items you want to arrange. This formula calculates the number of different ways you can arrange \(r\) items from \(n\) total items.

• \(P(10, 3)\) demonstrates arranging 3 items out of 10.

By dividing the factorial of \(n\) by the factorial of \(n-r\), we account for the different arrangements possible by just focusing on the \(r\) selected items.

Combinatorics

Combinatorics is a branch of mathematics concerning the counting, arrangement, and combination of objects. It allows us to solve problems related to predicting potential outcomes and finding patterns.
One aspect of combinatorics is permutations, which deal with ordered arrangements. Another is combinations, where the order does not matter, and is often denoted as \(C(n, r)\). While permutations focus on "arranging", combinations focus on "choosing."
Combinatorics answers questions like:

• In how many ways can we arrange a bookshelf with certain books?
• How many different committee groups can be formed from a pool of people?

By mastering combinatorics, you'll be equipped to tackle a wide range of mathematical and real-world scenarios where making predictions and arrangements are needed.