Problem 79 - Algebra 2

Question

Evaluate each expression. \(_{4} \mathrm{P}_{4}\)

Step-by-Step Solution

Answer

The permutation of 4 items taken 4 at a time \(_{4} \mathrm{P}_{4}\) is 24.

Step 1: Identify the values of n and r

From the given exercise, it's clear that both \(n\) and \(r\) are 4. This is represented as \(_{4} \mathrm{P}_{4}\)

Step 2: Apply the formula

Substitute the values into the formula to find the permutation. The formula for permutation is \(_{n} \mathrm{P}_{r} = \frac{n!}{(n-r)!}\). When both \(n\) and \(r\) are 4, the formula becomes \(_{4} \mathrm{P}_{4} = \frac{4!}{(4-4)!}\)

Step 3: Simplifying the Factorials

Now, simplify the factorials in the formula. \(4!= 4 \times 3 \times 2 \times 1\). Since \(4-4 = 0\), and \(0! = 1\) (by definition), the formula now becomes \(_{4} \mathrm{P}_{4} = \frac{4 \times 3 \times 2 \times 1}{1}\)

Step 4: Find the Permutation

Finally, simplify the formula to obtain the permutation. \(_{4} \mathrm{P}_{4} = 4 \times 3 \times 2 \times 1 = 24\)

Key Concepts

Factorials

The concept of factorials is key to understanding permutations. A factorial of a non-negative integer, denoted by an exclamation mark "!", is the product of all positive integers up to that number. It is represented mathematically as:

• For any positive integer \( n \), the factorial \( n! \) is defined as: \( n! = n \times (n-1) \times (n-2) \times \ldots \times 1 \).
• So, the factorial of 4, \( 4! \), is calculated as: \( 4 \times 3 \times 2 \times 1 = 24 \).
• By definition, \( 0! = 1 \). This may initially seem strange, but it simplifies many mathematical expressions and is a fundamental part of combinatorics.

Factorials grow extremely fast with larger numbers. Thus, it's essential to understand how they influence permutations and combinations typically used in counting problems.

Combinatorics

Combinatorics is the branch of mathematics concerned with counting, arranging, and studying patterns. It deals with both the theory and applications of how to count or arrange different elements. Combinatorial methods are used to calculate probabilities and solve problems in various domains like computer science, logic, and even biology.

• One of the primary objectives in combinatorics is to understand the different ways elements can be arranged or combined using permutations and combinations.
• When dealing with permutations, combinatorics often employ factorials for calculating arrangements when order matters.
• In contrast, combinations deal with selecting items where the order doesn't matter.

Combinatorics provides useful techniques for counting and arrangement problems, offering insight into a wide range of practical applications.

Permutation Formula

The permutation formula is a crucial tool in combinatorics, specifically when order matters in a selection of items. It helps determine how many ways a particular set of elements can be arranged.

• The general permutation formula is given by: \(_{n} \mathrm{P}_{r} = \frac{n!}{(n-r)!}\), where \( n \) is the total number of items, and \( r \) is the number of items to arrange.
• In this formula, \( n! \) calculates the total arrangements possible if all items were used, while dividing by \( (n-r)! \) removes the arrangements involving items not selected.
• For example, \(_{4} \mathrm{P}_{4} = \frac{4!}{(4-4)!} = \frac{24}{1} = 24\). Here, since all items are being used, the factorial simply provides the total number of permutations.

Understanding the permutation formula and its application helps solve various problems in permutations where sequence and arrangement are critical.