Problem 36 - College Algebra with Modeling and Visualization

Question

Evaluate the expression. \(P(12,3)\)

Step-by-Step Solution

Answer

The value of \( P(12, 3) \) is 1320.

Step 1: Understanding the Concept of Permutation

Permutation is used to determine the number of ways to arrange 'r' elements out of 'n' distinct elements. The formula for permutation is given by: \[ P(n, r) = \frac{n!}{(n-r)!} \] where 'n!' denotes the factorial of n.

Step 2: Substituting the Values into the Permutation Formula

We need to find \( P(12, 3) \). Substitute 'n' as 12 and 'r' as 3 into the permutation formula.\[ P(12, 3) = \frac{12!}{(12-3)!} \] Simplify it to \[ P(12, 3) = \frac{12!}{9!} \]

Step 3: Simplifying the Factorials

To simplify \( \frac{12!}{9!} \), realize that:\[ 12! = 12 \times 11 \times 10 \times 9! \]Therefore, the 9! in the numerator and denominator cancel out, simplifying the expression to: \[ 12 \times 11 \times 10 \]

Step 4: Calculating the Permutation

Perform the multiplication for the remaining terms in the expression:\[ 12 \times 11 = 132 \]\[ 132 \times 10 = 1320 \] Thus, \( P(12, 3) = 1320 \).

Key Concepts

factorial

Factorial is a fundamental concept in mathematics, especially in topics related to permutations and combinations. It's denoted by an exclamation mark, such as \( n! \), which represents the product of all positive integers from 1 to \( n \). For example, \( 4! \) means \( 4 \times 3 \times 2 \times 1 = 24 \). This means you multiply the number by every whole number less than itself down to 1.
Factorials are crucial when calculating permutations, as they help determine in how many different ways a set of items can be arranged. The growth of factorial numbers is very rapid, making it an interesting area of study in mathematics.
\( 0! \) is a special case, by definition, it is equal to 1. This definition helps ensure that formulas involving factorials work properly even when zero elements are involved.

combinatorics

Combinatorics is a branch of mathematics dealing with combinations, permutations, and counting the structures of a set. It explores the ways in which a collection of objects can be arranged or selected.
When dealing with permutations, you're interested in the order of the elements. If you think of arranging books on a shelf, the order they sit in matters, which is a classic permutation problem. Combinatorics provides the guidelines to calculate these arrangements efficiently and correctly.
Permutations are calculated using the formula: \( P(n, r) = \frac{n!}{(n-r)!} \), where order is important. Here, 'n' stands for the total number of items, and 'r' represents the number of items to arrange. This calculation gives the total ways to order a subset of a larger set.

mathematical expressions

Mathematical expressions are combinations of numbers, operations, and variables that represent a value. In the context of permutations, such expressions need simplification to find a final numeric answer.
Simplifying an expression involves executing operations and reducing terms to their simplest form. For example, in permutation calculations like \( \frac{12!}{9!} \), you need to understand that the factorial notation involves multiplying ascending numbers.
By cancelling out similar terms in the numerator and denominator, mathematical expressions become easier to solve. In this case, \( 12 \times 11 \times 10 \) are the remaining terms after cancelling the \( 9! \) part, leading to the solution.