Problem 13 - Algebra and Trigonometry

Question

Find the value of each permutation. $$ P(8,4) $$

Step-by-Step Solution

Answer

1680

Step 1: Understand the Permutation Formula

The formula for permutations is given by: \[ P(n, k) = \frac{n!}{(n - k)!} \] where \( n \) is the total number of items and \( k \) is the number of items to choose.

Step 2: Identify the Values

In this exercise, \( n = 8 \) and \( k = 4 \). Substitute these values into the permutation formula.

Step 3: Substitute the Values into the Formula

Using the formula: \[ P(8, 4) = \frac{8!}{(8 - 4)!} = \frac{8!}{4!} \]

Step 4: Calculate Factorials

Calculate the factorials involved: \[ 8! = 8 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1 = 40320 \] and \[ 4! = 4 \times 3 \times 2 \times 1 = 24 \]

Step 5: Divide the Factorials

Compute \( \frac{8!}{4!} \): \[ P(8, 4) = \frac{40320}{24} = 1680 \]

Key Concepts

Permutation Formula

In combinatorics, permutations deal with the arrangement of items. The permutation formula is a mathematical method used to determine the possible ways to arrange a subset of items from a larger set. This formula is crucial in solving problems related to order and arrangement.

The permutation formula is given by:

\[ P(n, k) = \frac{n!}{(n - k)!} \]
Here, \( n \) is the total number of items and \( k \) is the number of items to choose and arrange. This formula helps you find how many different sequences can be formed by selecting \( k \) items from \( n \) items without repetition.

For example, in the problem \( P(8, 4) \), you need to find the number of ways to choose and arrange 4 items out of 8.

Factorial Calculation

To solve permutation problems, calculating factorials is essential.
Factorials are the product of all positive integers up to a given number. The notation for factorial is an exclamation mark (!). For example, 5! (read as 'five factorial') is:

\[ 5! = 5 \times 4 \times 3 \times 2 \times 1 = 120 \]

In the given exercise, you need to find the factorial of both 8 and 4.

The calculation for 8! is:

\[ 8! = 8 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1 = 40320 \]

Similarly, the calculation for 4! is:

\[ 4! = 4 \times 3 \times 2 \times 1 = 24 \]

These calculations form a crucial part of the final step in finding permutations.

Mathematics Problem Solving

Mathematics problem-solving often involves a series of steps. For permutation calculations, the steps are sequential and logical. It includes understanding the problem, identifying the correct formulas, substituting values, and performing arithmetic operations.

In the given problem, we followed these steps:

• Understand the permutation formula.
• Identify the given values (n and k).
• Substitute the values into the formula.
• Calculate the factorials.
• Divide the factorials to get the final result.

By breaking the problem into smaller, manageable parts, you can systematically arrive at the solution. This structured approach is vital for tackling various mathematical problems.

Combinatorics

Combinatorics is a branch of mathematics dealing with combinations, permutations, and counting.
It is fundamental for fields such as computer science, probability, and statistics.

When working with permutations, you are finding the different ways to arrange a subset of items within a set. Unlike combinations, where only the selection of items is important, permutations emphasize the order of arrangement.

To distinguish between combinations and permutations, remember:

• Permutations: Order matters. Use the formula \( P(n, k) = \frac{n!}{(n - k)!} \).


• Combinations: Order does not matter. Use the formula \( C(n, k) = \frac{n!}{k!(n - k)!} \).

Understanding these basic principles of combinatorics will help you tackle more complex problems in mathematics and related fields.