Problem 5 - Algebra and Trigonometry

Question

Evaluate the expression. $$ P(8,3) $$

Step-by-Step Solution

Answer

The expression \(P(8, 3)\) evaluates to 336.

Step 1: Understand Permutation Notation

The notation \(P(n, r)\) represents the number of permutations of \(n\) items taken \(r\) at a time. It is calculated using the formula \(P(n, r) = \frac{n!}{(n-r)!}\). In this problem, \(n = 8\) and \(r = 3\).

Step 2: Calculate the Factorial of n

First, calculate \(8!\), which is the factorial of 8: \[8! = 8 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1 = 40320\].

Step 3: Calculate the Factorial of (n-r)

Calculate \((8-3)!\), which is \(5!\): \[5! = 5 \times 4 \times 3 \times 2 \times 1 = 120\].

Step 4: Apply the Permutation Formula

Substitute the values obtained into the permutation formula: \[P(8, 3) = \frac{8!}{(8-3)!} = \frac{40320}{120}\].

Step 5: Simplify the Expression

Perform the division to simplify \(\frac{40320}{120}\): \[\frac{40320}{120} = 336\]. Therefore, \(P(8, 3) = 336\).

Key Concepts

Factorial

The concept of a factorial is crucial when dealing with permutations. Essentially, factorial represents the product of all positive integers up to a specified number. Represented by an exclamation point (!), factorials grow surprisingly quickly as the number increases.

For example:

• \(1! = 1\)
• \(2! = 2 \times 1 = 2\)
• \(3! = 3 \times 2 \times 1 = 6\)
• \(4! = 4 \times 3 \times 2 \times 1 = 24\)

This goes on infinitely, following the formula \(n! = n \times (n-1) \times (n-2) \times ... \times 1\).

In the exercise, calculating both \(8!\) and \(5!\) demonstrated how evidence of factorial calculations simplifies the permutation formula. The use of factorials simplifies expressions by converting long multiplications into
simpler, more familiar terms.

Permutation Formula

Permutations are mathematical expressions dealing with the arrangement of items. Understanding the permutation formula is vital when you have distinct items and need to know how many
different ways you can arrange a subset of them.

The permutation formula is given by:\[P(n, r) = \frac{n!}{(n-r)!}\]where:

• \(n\) is the total number of items.
• \(r\) is the number of items to be chosen.

This formula allows you to find out how many different ways you can arrange \(r\) items out of a total of \(n\).

For example, in this exercise, \(P(8, 3)\) helps us understand how many ways we can arrange 3 items out of 8 by reducing the factorial calculations through division, to simplify complex
combinatorial problems. Understanding this formula is fundamental to solving permutation problems accurately.

Permutation Calculation

Calculating permutations involves applying the permutation formula to discover the number of possible arrangements. It combines understanding factorial calculations with division to
simplify and arrive at the number of unique sequences.

In the exercise, once you know how to compute factorials as \(8!\) and \(5!\), you simply use the formula \(P(n, r) = \frac{n!}{(n-r)!}\) with these values.

• First, calculate the factorial for the total number of items, \(n! = 8! = 40320\).
• Then, calculate the factorial for the difference, \((n-r)! = (8-3)! = 5! = 120\).

Finally, substitute these into the formula to get:\[P(8, 3) = \frac{40320}{120} = 336\].Therefore, there are 336 unique ways to arrange 3 items out of a total of 8. This showcases how permutations effectively organize and solve arrangements by using factorials and division.