Problem 9 - Algebra and Trigonometry

Question

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

Step-by-Step Solution

Answer

24

Step 1: Understand the Permutation Formula

The permutation formula for finding the number of ways to arrange r elements out of n is given by: \[ P(n, r) = \frac{n!}{(n - r)!} \].

Step 2: Substitute Values

Substitute the given values into the formula. Here, n = 4 and r = 4: \[ P(4, 4) = \frac{4!}{(4 - 4)!} \].

Step 3: Simplify the Factorial Expressions

Calculate the factorial expressions: \[ 4! = 4 \times 3 \times 2 \times 1 = 24 \] and \[ (4 - 4)! = 0! = 1 \] since the factorial of zero is 1.

Step 4: Divide the Values

Divide the factorials obtained in Step 3: \[ P(4, 4) = \frac{24}{1} = 24 \].

Key Concepts

Factorials

Factorials are fundamental in mathematics, especially in permutations and combinatorics. The factorial of a number, denoted as \({n!}\), is the product of all positive integers up to that number. For example, \({4!}\) is calculated as \({4 \times 3 \times 2 \times 1 = 24}\). Factorials grow very fast with the increase of the number. It's crucial to understand that \({0!}\) is defined as \({1}\), which is a unique property and often used in various mathematical expressions. Here’s a quick way to visualize some common factorial values:

• 1!=1
• 2!=2
• 3!=6
• 4!=24

Knowing how to compute factorials manually helps in solving more complex permutation problems.

Permutation Formula

Permutations involve arranging a set of elements in a specific sequence. The permutation formula is used to determine the number of ways to arrange 'r' elements from a set of 'n' elements. It is given by: \[ P(n, r) = \frac{n!}{(n - r)!} \] In this formula, \({n!}\) represents the factorial of the total elements, and \({(n - r)!}\) represents the factorial of the difference between the total elements and the number of elements to arrange. For instance, when \({P(4, 4)}\) is calculated, it fits the formula: \[ P(4, 4) = \frac{4!}{(4-4)!} = \frac{24}{1} = 24 \] This calculation shows there are 24 ways to arrange 4 elements out of 4.

Combinatorics

Combinatorics is a field of mathematics focused on counting, arrangement, and combination of objects. It involves principles like permutations, combinations, and the pigeonhole principle. Permutations, a common topic within combinatorics, deal with different ways of arranging a set of objects, where order matters. On the other hand, combinations focus on selecting items from a set without considering the order. To distinguish:

• Permutations: Order matters (e.g., different seating arrangements).
• Combinations: Order does not matter (e.g., selecting team members).

By understanding combinatorics, you can solve various complex problems by breaking them down into simpler counting problems. This makes it a vital tool in probability, computer science, and even everyday problem-solving.