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Chapter 5: Problem 10

Evaluate each expression. $$ \frac{6 ! 2 !}{4 !} $$

Short Answer

Expert verified
The expression evaluates to 60.

Step by step solution

01

- Understand Factorials

A factorial, denoted by the symbol (!), is the product of all positive integers up to a given number. For instance, 6! = 6 * 5 * 4 * 3 * 2 * 1.
02

- Compute Individual Factorials

Evaluate 6!, 2!, and 4! separately.\[6! = 6 \times 5 \times 4 \times 3 \times 2 \times 1 = 720\] \[2! = 2 \times 1 = 2\] \[4! = 4 \times 3 \times 2 \times 1 = 24\]
03

- Plug in the Values

Substitute the computed factorial values into the original expression. \[ \frac{6! \times 2!}{4!} \] becomes \[ \frac{720 \times 2}{24} \]
04

- Simplify the Expression

Perform the multiplication and division. \[ \frac{1440}{24} = 60\]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Mathematical Operations
Mathematical operations are the basic building blocks for all calculations. In this exercise, we see three main operations: multiplication, division, and the use of factorials. Multiplication combines numbers to find their total, while division splits a number into equal parts. The factorial operation, represented by the (!), is a more specialized operation. Understanding these basic operations is crucial because they help us simplify and solve expressions.
Factorial Notation
Factorial notation (denoted as !) is used in mathematics to indicate the product of all positive integers up to a certain number. For example, 4! means 4 x 3 x 2 x 1. Each factorial gets larger quickly as the number grows. Here is a breakdown of the factorials used in the problem:
  • 6! = 720 (which is 6 x 5 x 4 x 3 x 2 x 1)
  • 2! = 2 (which is 2 x 1)
  • 4! = 24 (which is 4 x 3 x 2 x 1)

Understanding this notation helps us to plug in these values accurately in an expression.
Simplifying Expressions
Simplifying expressions means breaking down complex calculations step by step. This can include following the order of operations (PEMDAS/BODMAS), simplifying fractions, or reducing terms. In our exercise, we first evaluate each factorial separately. Then, we substitute these values back into the original expression before performing the final division.
  • First, compute 6!, 2!, and 4! separately.
  • Next, substitute their values into the expression \( \frac{6! \times 2!}{4!} \)
  • Finally, simplify the fraction \( \frac{720 \times 2}{24} = 60 \)
By following these steps, we ensure that every operation is performed correctly, leading to the accurate solution.

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Most popular questions from this chapter

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