Problem 47 - Algebra for College Students

Question

Evaluate expression. \(\frac{P(12,3)}{3 !}\)

Step-by-Step Solution

Answer

220

Step 1: Understand the Expression

The expression involves a permutation and factorial operation. We need to evaluate the permutation first and then divide by the factorial.

Step 2: Recall the Permutation Formula

The formula for permutations is given by: \[ P(n, r) = \frac{n!}{(n-r)!} \]For our problem, we have \( P(12, 3) \).

Step 3: Calculate the Permutation

Substitute \( n = 12 \) and \( r = 3 \) into the permutation formula: \[ P(12, 3) = \frac{12!}{(12-3)!} = \frac{12!}{9!} \]Expand only the necessary part of the factorials: \[ P(12, 3) = 12 \times 11 \times 10 \]

Step 4: Calculate the Permutation Value

Perform the multiplication: \[ 12 \times 11 \times 10 = 1320 \]So, \( P(12, 3) = 1320 \).

Step 5: Calculate the Denominator (3!)

Factorial of 3 is given by: \( 3! = 3 \times 2 \times 1 = 6 \).

Step 6: Divide the Permutation by the Factorial

Divide the result of the permutation by the factorial: \[ \frac{P(12, 3)}{3!} = \frac{1320}{6} = 220 \]

Key Concepts

Factorials

Factorials are fundamental in various mathematical fields, especially in permutations and combinations. The factorial of a number, represented as ! (n!), is the product of all positive integers up to that number. For example, \(5! = 5 \times 4 \times 3 \times 2 \times 1 = 120\). It’s a way to count how many ways certain items can be ordered or selected. Factorials grow very quickly as the number increases, which is why they’re so powerful in combinatorics.

Combinatorics

Combinatorics is the branch of mathematics dealing with counting. It includes studying permutations and combinations. Permutations consider the order of elements, whereas combinations do not. For instance, if you have three letters A, B, C, the permutations are ABC, ACB, BAC, BCA, CAB, and CBA. Combinatorics helps solve problems related to counting possibilities, arrangements, and distributions.

Algebraic Expressions

Algebraic expressions are mathematical phrases that can contain numbers, variables, and operations. They are crucial for forming equations and solving problems. For example, in the permutation formula \(P(n, r) = \frac{n!}{(n-r)!}\), n and r are variables. Algebraic expressions allow us to generalize problems and solve them efficiently. Using them, we can simplify and manipulate mathematical statements to find solutions.

Order of Operations

The order of operations is essential to correctly solve mathematical expressions. It is often remembered by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division (left to right), Addition and Subtraction (left to right)). This standard ensures that everyone solves the expressions in the same way. When evaluating \( \frac{P(12, 3)}{3!}\), we first calculate the permutation \(P(12, 3)\), then the factorial \(3!\), and finally divide them, following the correct order of operations.