Problem 7 - Algebra and Trigonometry

Question

Use the formula for \(_{n} P_{r}\) to evaluate each expression. \(_{8} P_{0}\)

Step-by-Step Solution

Answer

The evaluated result of the expression _{8} P_{0} is 1.

Step 1: Identifying the values of n and r

In the expression _{8} P_{0}, n = 8 and r = 0. The values of n and r are identified from the given problem instance.

Step 2: Apply the permutation formula

The permutation formula is _{n} P_{r} = n! / (n-r)!. Substitute n = 8 and r = 0 in this formula.

Step 3: Simple calculation

Now, calculate it as _{8} P_{0} = 8! / (8-0)! = 8! / 8!. Since the denominator and numerator are the same, the result is 1

Key Concepts

Factorial

A factorial, denoted by an exclamation mark "!", is a mathematical operation that multiplies a series of descending natural numbers. For example, when you see "8!", it means 8 multiplied by all the integers below it down to 1, which is:

• 8! = 8 x 7 x 6 x 5 x 4 x 3 x 2 x 1 = 40,320

The concept of factorial is mostly used in permutations and combinations, allowing us to count the number of different ways to arrange or combine a set of items. When using factorials, it's essential to remember a few key rules:

• The factorial of 0 is always 1 (i.e., 0! = 1). This is because there is exactly one way to arrange zero items.
• Factorials grow really fast. Even calculating 10! gives a large number, 3,628,800.

Factorials play a crucial role in combinatorial and permutation formulas by providing a systematic method for arrangements.

Combinatorics

Combinatorics is a branch of mathematics that studies the different ways of arranging and combining objects. This field covers permutations, which consider order, and combinations, which do not. Combinatorics allows us to solve problems related to dice rolls, lottery probabilities, seating arrangements, and more. Some basic principles of combinatorics include:

• Permutations: Count the ways to arrange a subset of objects, where order matters.
• Combinations: Count the ways to choose objects where order does not matter.

A typical problem might involve calculating the number of ways to assign three tasks to four people. This can be approached using permutation formulas, leveraging the factorial operation to account for the different possible arrangements. In combinatorics, understanding the rules about order and repetition often guides selecting the right formula, whether permutation or combination.

Permutation Formula

The permutation formula is fundamental when dealing with arrangements where order is essential. The formula is expressed as \[ _{n}P_{r} = \frac{n!}{(n-r)!} \] Here, **n** is the total number of items, and **r** is the number of items to be arranged.

Why Use the Permutation Formula?

The permutation formula is essential when you need to determine how many ways you can organize a subset of items from a larger pool. For instance, if you have 8 books and want to know the number of ways to select and order 3 of them, you'd use this permutation formula.

Applying the Formula

Using the earlier example with the permutation \( _{8}P_{0} \): - First, identify that **n = 8** and **r = 0**.- Substitute these values into the permutation formula: \[ _{8}P_{0} = \frac{8!}{(8-0)!} = \frac{8!}{8!} \] Since both the numerator and denominator are the same (8!), the result is 1. That means there's only one way to arrange zero items out of eight, which aligns with the factorial rule that 0! equals 1. In summary, the permutation formula simplifies calculating arrangements by handling complexity with factorial operations.