Problem 18 - College Algebra

Question

For the following exercises, compute the value of the expression. $$ P(9,6) $$

Step-by-Step Solution

Answer

The value of \( P(9, 6) \) is 60,480.

Step 1: Understand the Notation

The notation \( P(n, r) \) represents a permutation where we want to choose \( r \) objects from \( n \) total objects, considering the order. It is calculated using the formula \( P(n, r) = \frac{n!}{(n-r)!} \).

Step 2: Identify the Values of n and r

For the expression \( P(9, 6) \), \( n = 9 \) and \( r = 6 \). We need to calculate the number of permutations where 6 objects are chosen from a set of 9 objects.

Step 3: Calculate n-r

Subtract \( r \) from \( n \) to get \( n-r = 9-6 = 3 \). This will help us in the calculation of the factorials needed for the permutation formula.

Step 4: Compute Factorials

Compute the factorial of \( n \) and \( n-r \):- \( 9! = 9 \times 8 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1 = 362880 \)- \( (9-6)! = 3! = 3 \times 2 \times 1 = 6 \)

Step 5: Apply the Permutation Formula

Using the formula \( P(n, r) = \frac{n!}{(n-r)!} \), substitute the calculated factorials:\[ P(9, 6) = \frac{9!}{3!} = \frac{362880}{6} = 60480 \]

Step 6: Conclusion

The result from the above calculation shows that there are 60,480 different permutations of choosing 6 objects from a total of 9, considering the order.

Key Concepts

Factorial

Understanding the concept of a factorial is crucial when working with permutations. A factorial, denoted as \(n!\), is the product of all positive integers up to a chosen number \(n\). For example, \(5!\) (read as "five factorial") is computed as \(5 \times 4 \times 3 \times 2 \times 1 = 120\).
To solve permutation problems like \(P(9, 6)\), you need to calculate factorials such as \(9!\) and \(3!\). Here:

• \(9! = 9 \times 8 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1 = 362,880\)
• \(3! = 3 \times 2 \times 1 = 6\)

Remember, the factorial of any number is a recursive multiplication down to 1. This concept helps in simplifying and calculating bigger equations in combinatorics.

Combinatorics

Combinatorics is a branch of mathematics that deals with counting, both as a means and an end in obtaining some measure of structure. One of its primary concerns is understanding permutation and combination.
In the case of permutations, you're determining all possible arrangements where order counts. For instance, in \(P(9, 6)\), you're exploring how many ways you can choose 6 objects from 9, where the position matters.
The key formula here is the permutation formula, \(P(n, r) = \frac{n!}{(n-r)!}\), which uses factorials to calculate these arrangements.
Combinatorics is applied not only in mathematics but in computer science, economics, and even biology, wherever determining possibilities, organization, and configurations is required.

Mathematical Notation

Mathematical notation serves as a universal language for mathematicians and students. It provides a compact and uniform way to express mathematical ideas.
The notation \(P(n, r)\) specifically denotes the number of permutations of \(n\) objects taken \(r\) at a time, where order is important.
This is just one example of how mathematical notation simplifies expressions that might otherwise require lengthy descriptions.
Using such quick representations allows for complex calculations to be understood and communicated efficiently across different fields.
Having a strong grasp over mathematical notation will aid in dissecting and solving problems both for academic purposes and real-life applications.