Question

Show that \(4 !+3 ! \neq(4+3) !\)

Short answer

4! + 3! = 30 and (4+3)! = 5040, thus they are not equal.

Step-by-step solution

Understand Factorials

Factorials are the product of all positive integers up to a given number. For example, the factorial of 4, written as 4!, is calculated as 4 × 3 × 2 × 1.

Calculate 4!

Calculate the factorial of 4: 4! = 4 × 3 × 2 × 1 = 24

Calculate 3!

Calculate the factorial of 3: 3! = 3 × 2 × 1 = 6

Add 4! and 3!

Add the values of 4! and 3!: 4! + 3! = 24 + 6 = 30

Calculate (4+3)!

Calculate the factorial of (4+3), which is 7! : 7! = 7 × 6 × 5 × 4 × 3 × 2 × 1 = 5040

Compare the Results

Compare the results of 4! + 3! and (4+3)! : 30 eq 5040

Conclusion

Therefore, 4! + 3! eq (4+3)!

Key concepts

Factorial Calculation

Factorials are an essential concept in mathematics, particularly in algebra and calculus. A factorial, denoted as !, is the product of all positive integers up to a given number. It is written as n!, where n is a positive integer. For example, the factorial of 4 (4!) is calculated by multiplying 4 × 3 × 2 × 1, which equals 24.

Factorials grow very quickly as the numbers get larger. That's because each new number multiplies the product of all the previous numbers. As a student, mastering factorial calculation will help you understand more complex mathematical topics. Here’s a quick reference to calculate some basic factorials:

• 0! = 1 (by definition)
• 1! = 1
• 2! = 2 × 1 = 2
• 3! = 3 × 2 × 1 = 6
• 4! = 4 × 3 × 2 × 1 = 24

Understanding how to calculate factorials is a stepping stone to comprehending and solving more intricate problems in mathematics.

Comparing Factorials

Comparing factorials involves evaluating and understanding the different results you get from factorial operations. In this exercise, we specifically compare two different expressions involving factorials: 4! + 3! and (4+3)!.

First, let's break down what each expression means:

• 4! is the factorial of 4.
• 3! is the factorial of 3.
• (4+3)! is the factorial of the sum of 4 and 3, which is 7.

Now, calculate the individual factorials:
4! = 4 × 3 × 2 × 1 = 24
3! = 3 × 2 × 1 = 6
Adding these, we get: 4! + 3! = 24 + 6 = 30

Next, calculate 7! because (4+3) = 7:
7! = 7 × 6 × 5 × 4 × 3 × 2 × 1 = 5040

As we can see, 30 (from 4! + 3!) is not equal to 5040 (from 7!). This illustrates that adding factorials and taking the factorial of a sum are two entirely different operations with vastly different outcomes. Understanding this difference is crucial, especially in higher mathematics and algebra.

Elementary Algebra

Elementary algebra is the branch of mathematics that starts dealing with variable expressions and arithmetic operations. It includes understanding and using basic algebraic concepts such as variables, constants, coefficients, equations, and functions.

Factorials play a noticeable role in elementary algebra because they often appear in permutations, combinations, and various algebraic expressions. The exercise of showing that 4! + 3! ≠ (4+3)! reinforces a few critical algebraic principles:

• Operators and their precedence: Factorial ( !) has priority and is evaluated before addition or multiplication of expressions.
• Function operations: Understanding when to sum functions versus applying functions (like factorial) to sums.
• Comparative analysis: Analyzing and comparing outcomes of different algebraic processes.

Algebra, while it can seem complex, builds from these fundamental concepts. Grasping the basic principles of operations like factorials in elementary algebra allows you to solve more intricate mathematical problems and prepares you for advanced studies in mathematics. Always remember to break problems down into smaller steps to understand each part clearly.