Problem 7 - Algebra and Trigonometry

Question

In Problems \(7-14\), find the value of each permutation. $$ P(6,2) $$

Step-by-Step Solution

Answer

The value of P(6, 2) is 30.

Step 1 - Understand the Permutation Formula

The general formula for permutations 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 arrange.

Step 2 - Substitute Given Values into the Formula

Given \( n = 6 \) and \( r = 2 \), substitute these values into the permutation formula:\[ P(6, 2) = \frac{6!}{(6-2)!} \]

Step 3 - Simplify the Factorials

Calculate the factorials involved. First, find 6!:\[ 6! = 6 \times 5 \times 4 \times 3 \times 2 \times 1 = 720 \]Next, find (6-2)!:\[ 4! = 4 \times 3 \times 2 \times 1 = 24 \]Now, substitute these values back into the formula:

Step 4 - Calculate the Permutation

Divide 6! by 4! as follows:\[ P(6, 2) = \frac{720}{24} = 30 \]

Key Concepts

factorials

A factorial, represented by the symbol \(!\), is a mathematical function that multiplies a number by every positive whole number less than it. For example, the factorial of 5, written as \(!5\), is calculated as: \[ 5! = 5 \times 4 \times 3 \times 2 \times 1 = 120. \] Factorials are used extensively in permutations, combinations, and other areas of combinatorics. They grow rapidly with increasing numbers. For instance, 10! is 3,628,800, far larger than many might expect from simply multiplying 1 through 10. It's essential to become comfortable with calculating factorials for solving problems involving arrangements and selections.

combinatorics

Combinatorics is the area of mathematics focused on counting, arrangement, and combination of objects. It's a foundational aspect of discrete mathematics, where the objects are considered distinct and separate. Within combinatorics, the study of permutations deals with the different ways to arrange a certain number of items. There are also combinations, which address ways to choose items without concern for order. Practical applications are widespread, from solving puzzles to analyzing probabilities in games or working with data structures in computer science.

permutation formula

The permutation formula is used to determine the number of ways to arrange a subset of items from a larger set. Mathematically, it's expressed as: \[ P(n, r) = \frac{n!}{(n-r)!} \] Here, \(n\) represents the total number of items, and \(r\) represents the number of items being arranged. For example, in the given problem where \(n = 6\) and \(r = 2\), the permutation formula \(P(6, 2)\) is calculated as: \[ P(6, 2) = \frac{6!}{4!} = \frac{720}{24} = 30 \] This tells us there are 30 ways to arrange 2 items out of 6.

arrangement of items

Permutations specifically deal with the arrangement of items where order matters. When arranging items, every distinct sequence counts as a different permutation. For instance, with items \('A', 'B'\) and \('C'\), arranging them in 2 slots could yield permutations like \('AB', 'BA', 'AC', 'CA', 'BC', 'CB'\). Each unique sequence is a valid permutation. Using the permutation formula, we can efficiently calculate the total number of such arrangements without needing to list each possibility manually.

mathematical calculations

Mathematical calculations for permutations often involve factorials, as seen in the formula \(P(n, r) = \frac{n!}{(n-r)!}\). A step-by-step approach simplifies complex problems significantly. First, determine the values of \(n\) and \(r\). Next, calculate the factorials \(n!\) and \((n-r)!\). Finally, perform the division specified by the formula. For accuracy, breaking the problem into smaller parts or using computational tools can help manage large numbers typical in factorials. Besides permutations, such factorial-based calculations also appear in combinations, probability theory, and other mathematical analyses.