Problem 39 - Algebra 2

Question

What is the value of \(\frac{6 !}{2 !} ?\) A. 3 B. 60 C. 360 D. 720

Step-by-Step Solution

Answer

The value is 360 (Option C).

Step 1: Understand the Problem

We are asked to find the value of the expression \(\frac{6!}{2!}\). In this expression, \(!\) denotes factorial, which means the product of all positive integers up to that number. Our task is to simplify and compute this expression.

Step 2: Calculate 6-Factorial

Calculate \(6!\), which is the product of all positive integers up to 6:\[6! = 6 \times 5 \times 4 \times 3 \times 2 \times 1 = 720\]

Step 3: Calculate 2-Factorial

Calculate \(2!\), which is the product of all positive integers up to 2:\[2! = 2 \times 1 = 2\]

Step 4: Divide the Factorials

Now that we have both factorials, we need to divide \(6!\) by \(2!\):\[\frac{6!}{2!} = \frac{720}{2} = 360\]

Step 5: Choose the Correct Answer

After performing the division, we find that \(\frac{720}{2} = 360\). Therefore, the correct answer from the given options is C: 360.

Key Concepts

Algebra

Algebra is like the language of mathematics. It allows us to describe relationships and changes using symbols and letters. In algebra, we often use letters to represent numbers, helping us to solve equations and understand patterns. For instance, when we deal with a factorial like \(6!\), algebra provides a way to simplify and manipulate such expressions easily. This approach aids in quick calculations or finding unknown values in an equation.
Algebra also helps organize our problem-solving. By using rules and operations, we can work through complex expressions systematically. For example, understanding the order of operations and how to deal with parentheses are fundamental algebraic skills. Applying these concepts to factorials enables us to solve problems like \(\frac{6!}{2!}\) efficiently.

Factorial Simplification

Factorial simplification is about breaking down or reducing factorial expressions to make them easier to work with. A factorial, represented by \(n!\), involves multiplying all whole numbers from 1 up to \(n\). Simplifying factorials can help us handle large numbers in a more concise form.
When approaching a problem like \(\frac{6!}{2!}\), the goal is to simplify the expression first by calculating each factorial individually:

• Calculate \(6! = 6 \times 5 \times 4 \times 3 \times 2 \times 1 = 720\)
• Calculate \(2! = 2 \times 1 = 2\)

Once the factorials are calculated, simplification of the expression through division becomes straightforward. This kind of simplification not only makes the arithmetic manageable but also deepens our understanding of the underlying mathematical relationships.
Recognizing how to break down and simplify factorials is crucial in many mathematical contexts, from probability to algebra, allowing us to tackle more complex problems seamlessly.

Division of Factorials

The division of factorials is a mathematical operation that reduces the size of expressions by canceling out common terms. This operation is critical when dealing with large factorial numbers because it can vastly simplify calculations.
In the expression \(\frac{6!}{2!}\):

• First calculate the individual factorials: \(6! = 720\) and \(2! = 2\).
• Then compute the division: \(\frac{720}{2} = 360\).

When dividing factorials, it's often useful to recognize patterns or simplifications early. For instance, recognizing that many terms in \(6!\) and \(2!\) overlap can allow for quick cancelation of terms without expanding everything fully, though in this case, direct calculation works well due to small numbers.
Overall, learning to efficiently divide factorials can be very advantageous, especially in fields like combinatorics and calculus, where factorials are common. This skill will streamline calculations and help comprehend more advanced mathematical concepts.