Problem 11 - Algebra 2

Question

Evaluate each expression. \(P(8,2)\)

Step-by-Step Solution

Answer

The value of \(P(8,2)\) is 56.

Step 1: Understanding Permutations

The expression \(P(8, 2)\) refers to the number of permutations of 8 items taken 2 at a time. In permutations, the order matters. The formula for finding permutations \(P(n, r)\) is given by: \[ P(n, r) = \frac{n!}{(n-r)!} \] where \(n\) is the total number of items, and \(r\) is the number of items to be arranged.

Step 2: Substitute the Values into the Formula

Using the permutation formula \( P(n, r) = \frac{n!}{(n-r)!} \), substitute \(n = 8\) and \(r = 2\). This gives us: \[ P(8, 2) = \frac{8!}{(8-2)!} = \frac{8!}{6!} \]

Step 3: Calculate Factorials and Simplify

Calculate the factorials: \(8! = 8 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1\) and \(6! = 6 \times 5 \times 4 \times 3 \times 2 \times 1\). Divide \(8!\) by \(6!\): \[ \frac{8!}{6!} = \frac{8 \times 7 \times 6!}{6!} = 8 \times 7 \] which simplifies to 56.

Key Concepts

Factorials

Factorials are a fundamental concept in mathematics, especially in topics like permutations and combinations. When a number is followed by an exclamation point, like in "8!", it means you need to calculate the factorial of that number. To find the factorial of a number, multiply it by all the positive integers less than it. For example, to find 8!, you multiply like this:

• 8 × 7 × 6 × 5 × 4 × 3 × 2 × 1

This product equals 40,320. Factorials are essential because they represent the total number of ways to arrange a set number of items. An important thing to note is that 0! is defined to be 1. This unique case is useful in various mathematical functions and formulas, providing a foundation for problems involving permutations.

Arrangement

The concept of arrangement in mathematics often refers to permutations, where the order of objects is crucial. In permutations, rearranging the same set of items results in different outcomes. For example, in the problem of finding the permutations of 8 items taken 2 at a time, you can select and order any two from eight items. Since the sequence in which you arrange the items matters, calculating different arrangements means you can select a different first item, which affects the rest. The increased number of possibilities makes the calculation of arrangements very useful.

• This concept applies to organizing competitions, seating arrangements, and more.
• Unlike combinations, arrangement is all about finding different sequences.

Understanding arrangement means recognizing each choice leads to a distinct sequence, enhancing the complexity and variety in outcomes.

Permutation Formula

The permutation formula is essential for calculating the number of possible arrangements where the order of selection is important. The specific formula used is:

• \( P(n, r) = \frac{n!}{(n-r)!} \)

Here, \(n\) represents the total number of items available, and \(r\) is the number of items you want to arrange. This formula helps you find all possible sequences of a subset from a larger set without replacement.
In the example provided, \( P(8, 2) = \frac{8!}{6!} = 8 \times 7 \), resulting in 56 possible arrangements. It's useful to compare this with combinations where order does not matter, changing how problems are approached.
Using the permutation formula, you can solve various real-world problems related to organizational tasks and ordering processes. Knowing how to manipulate and apply this equation allows for efficient problem-solving in both theoretical and practical contexts.