Problem 46 - Algebra 2

Question

Evaluate each expression. \(\frac{4 !}{2 ! 2 !}\)

Step-by-Step Solution

Answer

The value of the expression is 6.

Step 1: Understand the Factorial Notation

Factorials are products of all positive integers up to a given number. For example, for any integer n, \( n! = n \times (n - 1) \times (n - 2) \times \ldots \times 1 \). In this problem, we will need to calculate \( 4! \), \( 2! \), and \( 2! \).

Step 2: Calculate the Factorial Values

Calculate \( 4! \), \( 2! \), and \( 2! \). - \( 4! = 4 \times 3 \times 2 \times 1 = 24 \) - \( 2! = 2 \times 1 = 2 \) - \( 2! = 2 \times 1 = 2 \)

Step 3: Substitute Factorial Values into the Expression

Replace the factorial values in the expression \( \frac{4!}{2!\times 2!} \) with the calculated results:\[ \frac{24}{2 \times 2} \]

Step 4: Simplify the Expression

Simplify the expression by performing the arithmetic operations:First compute \( 2 \times 2 = 4 \). Then compute \( \frac{24}{4} = 6 \).

Step 5: Conclude the Evaluation

After simplification, we find that the value of the expression \( \frac{4!}{2!\times 2!} \) is 6.

Key Concepts

Combinatorics

Combinatorics is a field of mathematics focused on counting, arranging, and analyzing configurations of sets. It often deals with determining the number of ways certain patterns can be formed. In our exercise, combinatorics is essential because we are evaluating an expression from a binomial coefficient, specifically \( \frac{4!}{2!\times 2!} \), which represents the combination formula \( \binom{4}{2} \). This helps determine how many ways we can choose 2 items out of 4. The expression thus evaluates a combinatorial scenario where repetition is not allowed.

Arithmetic Operations

Arithmetic operations are basic calculations such as addition, subtraction, multiplication, and division. They are necessary for solving most mathematical expressions. In the original exercise, after calculating the factorials, arithmetic operations help simplify the expression. Once the factorials are substituted, you perform division:

• First, calculate \( 2 \times 2 = 4 \).
• Then, simplify the expression \( \frac{24}{4} \) to get 6.

These operations are simple yet fundamental in arriving at the correct solution to the problem.

Permutations

Permutations pertain to arrangements or sequences formed by objects. Unlike combinations, permutations consider order. Although the original exercise is more about combinations than permutations, knowing the difference is crucial. In permutations, formulas often involve factorials. For example, arranging 4 objects in different sequences is calculated using \( 4! \), resulting in 24 different arrangements. While permutations calculate every possible order, combinations like in our exercise calculate unordered selections, hence the division by repeating \( 2! \). This prevents over-counting scenarios where order does not matter.

Mathematical Notation

Mathematical notation provides a universally accepted language to express ideas and calculations. It includes symbols like \(!\) for factorials and \(\frac{a}{b}\) for fractions. In our example:

• Factorial notation \( n! \) simplifies long multiplication sequences, such as \( 4! = 24 \).
• Fractions help in dividing quantities, evident in \( \frac{4!}{2!\times 2!} \).

Understanding notation allows for easier interpretation and solving of mathematical problems. Such notation encapsulates complex ideas in a standard format, crucial for mathematicians and students alike.