Problem 12 - Algebra 2

Question

Evaluate each expression. \(P(9,1)\)

Step-by-Step Solution

Answer

The value of \(P(9, 1)\) is 9.

Step 1: Understand the Permutation Formula

We are required to evaluate the expression \(P(9,1)\). The notation \(P(n, r)\) stands for the number of permutations of \(n\) items taken \(r\) at a time. The formula to calculate permutations is given by \[P(n, r) = \frac{n!}{(n-r)!}\]. In this problem, \(n = 9\) and \(r = 1\).

Step 2: Substitute Values into the Formula

Use the permutation formula \(P(n, r) = \frac{n!}{(n-r)!}\) with \(n = 9\) and \(r = 1\). Substitute these values into the formula to get: \[P(9, 1) = \frac{9!}{(9-1)!}\].

Step 3: Simplify the Denominator

Calculate \((n-r)!\) where \(n = 9\) and \(r = 1\). This gives \((9-1)! = 8!\). Therefore, the expression becomes: \[P(9, 1) = \frac{9!}{8!}\].

Step 4: Calculate the Factorial Values

Recall that \(n!\) means multiplying all whole numbers from 1 to \(n\). Thus, \(9! = 9 \times 8 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1\) and \(8! = 8 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1\).

Step 5: Simplify the Expression

Notice that \(8!\) cancels out from \(9!\) in the fraction \(\frac{9!}{8!}\). Therefore, \[P(9, 1) = 9\times \frac{8!}{8!} = 9\].

Step 6: Conclude the Solution

The simplified result of the permutation \(P(9, 1)\) is 9.

Key Concepts

Factorial

Factorials are fundamental in mathematics, especially in permutations and combinations. The factorial of a number, denoted as \(n!\), represents the product of all positive integers up to that number. For instance, the factorial of 5 (written as 5!) is calculated as \(5 \times 4 \times 3 \times 2 \times 1 = 120\).
Factorials grow exponentially, meaning they get very large very quickly. This explosive growth is due to multiplying a sequence of successively larger integers. It forms the basis of several mathematical formulas, including permutations and combinations.
The concept of a factorial is not just confined to whole numbers. In advanced mathematics, it extends to other areas through functions like the gamma function, which generalizes the factorial concept. However, in combinatorics, we deal with whole-number factorials.

Permutation Formula

The permutation formula allows us to calculate the number of possible arrangements of a set of items, where the order of arrangement matters. It is expressed as \[P(n, r) = \frac{n!}{(n-r)!}\].
This formula determines how many different ways we can choose and arrange 'r' items from a larger set of 'n' items. It is crucial to remember the distinction in permutations: the arrangement order affects the outcome.

• In permutations, \(n\) represents the total number of items available.
• The \(r\) stands for how many of those items we are choosing to permute or arrange.
• The result will tell us the number of unique orderings possible.

In practice, solving a permutation problem involves computing factorials, as shown in this exercise with \(P(9, 1)\). We learned how \(9!\) and \(8!\) come into play in our solution by canceling out common terms to simplify the problem and find the result.

Combinatorics

Combinatorics is a branch of mathematics that deals with counting, arrangement, and combination of objects. It's all about finding systematic ways to count configurations and understand structures.
Within combinatorics, permutations (as discussed in the previous section) and combinations are two of the most important concepts. While permutations consider order to be important, combinations ignore the sequence and only regard the selection of items.

• Combinatorics is used to solve problems related to probability, statistics, computer science, and more.
• It allows for efficient problem-solving in scenarios involving large datasets or multiple possibilities.
• Its tools include factorial functions, permutation and combination formulas, among others.

Combinatorics forms a mathematical toolkit that simplifies complex problems into manageable calculations, enabling us to find solutions in many fields of science and technology. By understanding permutations within combinatorics, we can solve real-world problems efficiently.