Problem 46 - Algebra for College Students

Question

Evaluate expression. \(\frac{P(8,3)}{3 !}\)

Step-by-Step Solution

Answer

The value of the expression is 56.

Step 1 - Calculate the Permutation

First, find the permutation of 8 objects taken 3 at a time, which is given by the formula for permutation: \[P(n, r) = \frac{n!}{(n-r)!}\] Substitute \(n = 8\) and \(r = 3\): \[P(8, 3) = \frac{8!}{(8-3)!} = \frac{8!}{5!}\]

Step 2 - Simplify the Permutation Expression

To simplify \(\frac{8!}{5!}\), recognize that \(8! = 8 \times 7 \times 6 \times 5!\). Thus, \[\frac{8!}{5!} = 8 \times 7 \times 6 = 336\] So, \(P(8, 3) = 336\).

Step 3 - Calculate the Factorial of 3

Next, calculate the factorial of 3: \[3! = 3 \times 2 \times 1 = 6\]

Step 4 - Divide the Permutation by 3!

Finally, divide the permutation value found in Step 2 by the factorial calculated in Step 3: \[\frac{P(8, 3)}{3!} = \frac{336}{6} = 56\]

Key Concepts

Factorials

In mathematics, a factorial is the product of all positive integers up to a given number. It's denoted with an exclamation mark (!) after the number. For instance, the factorial of 3 (written as 3!) is calculated as follows:

\[3! = 3 \times 2 \times 1 = 6\]

This means we multiply together all whole numbers from 1 up to 3. Factorials are crucial in permutations and combinations because they help determine the number of possible ways to arrange or select items.

Here are some examples to illustrate:

• 4! = 4 \times 3 \times 2 \times 1 = 24
• 5! = 5 \times 4 \times 3 \times 2 \times 1 = 120

Notice how quickly the value grows as the number increases! This rapid growth is an important characteristic of factorials.

Combinatorics

Combinatorics is a branch of mathematics that focuses on counting, arrangement, and combination of objects. It includes concepts like permutations and combinations, which help us figure out all possible ways to organize items.

Combinatorics is essential in fields such as computer science, cryptography, and probability. It helps solve problems related to arranging objects in specific orders or selecting groups of items effectively.

For example:

• In permutations, order matters. If you have a set of 3 letters {A, B, C} and you want to arrange them in different ways, the permutations would be ABC, ACB, BAC, BCA, CAB, CBA.
• In combinations, order does not matter. Using the same set {A, B, C}, if you want to pick 2 letters in various ways, the combinations would be AB, AC, and BC.

By understanding combinatorics, students can tackle a wide range of mathematical problems more effectively.

Permutation Formula

A permutation refers to a specific arrangement of items where the order is important. The permutation formula, \[P(n, r) = \frac{n!}{(n-r)!}\]is used to calculate the number of ways to arrange ‘r’ items out of ‘n’ options. Here, \(n\) represents the total number of items, and \(r\) represents the number of items being chosen.

Let’s look at an example:
To find the number of ways to arrange 3 items out of 8:

• Substitute \(n = 8\) and \(r = 3\)
• \[P(8, 3) = \frac{8!}{(8-3)!} = \frac{8!}{5!}\]

Simplifying further:

• \[8! = 8 \times 7 \times 6 \times 5!\]
• Thus, \(\frac{8!}{5!} = 8 \times 7 \times 6 = 336\)

So, the number of ways to arrange 3 items out of 8 is 336. This showcases the importance and applicability of the permutation formula in various scenarios like seating arrangements, schedules, and task orders.