Problem 2 - Algebra 2

Question

Evaluate each expression. \(P(6,3)\)

Step-by-Step Solution

Answer

The value of \(P(6, 3)\) is 120.

Step 1: Understand the Permutation Formula

Permutations are arrangements of objects where order matters. The formula for permutations is given by \(P(n, r) = \frac{n!}{(n-r)!}\) where \(n\) is the total number of objects, and \(r\) is the number of objects to choose.

Step 2: Identify the Given Values

From the expression \(P(6, 3)\), we have \(n = 6\) and \(r = 3\). This means we need to arrange 6 objects and select 3 of them.

Step 3: Calculate the Factorials

Calculate the factorial of \(n\) and \(n-r\). First, \(6! = 6 \times 5 \times 4 \times 3 \times 2 \times 1 = 720\). Second, \((6-3)! = 3! = 3 \times 2 \times 1 = 6\).

Step 4: Substitute Values into Permutation Formula

Substitute the values calculated in the permutation formula: \[P(6, 3) = \frac{6!}{(6-3)!} = \frac{720}{6}\].

Step 5: Simplify the Expression

Divide the factorial results: \(\frac{720}{6} = 120\). Hence, \(P(6, 3) = 120\).

Key Concepts

Understanding Factorials

Factorials are a fundamental mathematical concept used extensively in permutations, combinations, and other areas of mathematics. A factorial, denoted by an exclamation mark, is the product of all positive integers up to a certain number. For example, the factorial of 5, written as 5!, is given by:

• 5! = 5 × 4 × 3 × 2 × 1 = 120

Factorials grow very rapidly with increasing numbers. They are used to determine the number of ways to arrange or sequence a set of items. Because of this, they are a key part of the permutation formula, which allows us to calculate arrangements where order is important. Understanding how to compute factorials accurately is crucial for solving problems that involve permutations and combinations.
In the context of permutations, we often calculate the factorial of not only a total number of items but also a subset of these items, as seen in expressions like \(n-r\)! where \(n\) and \(r\) represent total and chosen items respectively.

Basics of Combinatorics

Combinatorics is the branch of mathematics dealing with combinations, permutations, and the counting of objects based on certain rules. It is widely used in fields like probability, statistics, and computer science.
Permutations, a major concept under combinatorics, involve arranging a set of items where order matters. The formula to find permutations for selecting \(r\) items from \(n\) total items is:

• \(P(n, r) = \frac{n!}{(n-r)!}\)

This formula captures how many possible orderings there are when choosing \(r\) items from \(n\). An important aspect is that in permutations, different orders of the same items are considered distinct arrangements.
In combinatorics, the distinction between permutations and combinations is crucial. While permutations count arrangements, combinations count selections where order does not matter. For any student exploring mathematical tasks involving ordering or selecting, a grasp of combinatorial methods is essential.

Formulating Mathematical Expressions

Mathematical expressions are used to represent numbers and operations in a symbolic form. They can range from simple operations, like addition or subtraction, to more complex structures involving factorials and permutation formulas.
Expressions like \(P(6, 3)\) serve to encapsulate a problem in a concise way using well-known formulas. Here, \(P(6, 3)\) is a symbolic representation of a permutation problem involving 6 items, from which 3 are to be chosen and arranged.

• This is calculated using the formula \(P(n, r) = \frac{n!}{(n-r)!}\).
• The expression implies that the result is derived by utilizing factorials, which are foundational to these types of calculations.

When working with mathematical expressions, it's important to understand the underlying principles each symbol represents, such as in \(P(6,3)\). This helps to unpack the expression, perform calculations accurately, and derive meaningful results.