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Chapter 11: Problem 23

Evaluate each factorial expression. \(7 !-3 !\)

Short Answer

Expert verified
The value of the given factorial expression \(7! - 3!\) is 5034.

Step by step solution

01

Evaluate the Factorial of 7

The factorial of 7 is the product of all positive integers from 1 to 7. Hence, \(7! = 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1 = 5040\).
02

Evaluate the Factorial of 3

The factorial of 3 is the product of all positive integers from 1 to 3. Hence, \(3! = 3 \times 2 \times 1 = 6\).
03

Subtract the Factorial of 3 From the Factorial of 7

Now to find the difference between the two factorials, we subtract the factorial of 3 from the factorial of 7. Hence, \(7! - 3! = 5040 - 6 = 5034\).

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Key Concepts

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

Factorial Notation
Factorial notation is a mathematical operation that finds widespread use in areas like permutations and combinations. A factorial is denoted by the exclamation mark following a number, such as in "7!". It represents the product of all positive integers up to and including the number itself. For instance, the factorial of 7, expressed as \(7!\), is calculated as follows:
  • Start with the number itself: 7.
  • Multiply it by every positive integer less than it: 6, 5, 4, 3, 2, and 1.
So, \(7! = 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1 = 5040\).
Thus, when someone uses factorial notation, they are indicating the use of multiplicative sequences building up to that number.
This operation is straightforward but can yield very large results quickly, even for relatively small numbers.
Subtraction of Factorials
Subtracting factorials involves taking the result of one factorial and subtracting the result of another. The subtraction process is simple arithmetic once the factorials are evaluated.
In our exercise, you are asked to find the difference between the factorial of 7 and the factorial of 3.
  • We first find \(7! = 5040\).
  • Next, we find \(3! = 6\).
  • Then, subtract \(3!\) from \(7!\) to get \(7! - 3! = 5040 - 6 = 5034\).
It is important to evaluate each factorial separately before performing the subtraction.
Factorials grow at an extremely rapid rate, so even subtracting a smaller factorial from a larger one can yield large differences.
This operation reinforces the need to be careful with arithmetic calculations involving factorials to avoid errors.
Basic Arithmetic Operations
Understanding basic arithmetic operations is crucial when working with factorials. These operations include addition, subtraction, multiplication, and division. In the context of this exercise, subtraction is the main focus.
Before performing any arithmetic operation, it is essential to evaluate the individual factorials fully.
Then, use the principles of these operations like subtraction:
  • Identify the terms. In our case, \(7!\) and \(3!\).
  • Perform the subtraction by taking the smaller result from the larger one.
  • Ensure accuracy by double-checking calculations to prevent early mistakes.
Mastering basic arithmetic provides a stable foundation for tackling more complex mathematical problems.
Practicing these operations regularly boosts confidence and skill in handling various calculations including those involving large numbers or factorials.

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