Problem 27 - College Algebra with Modeling and Visualization

Question

Evaluate the expression. \(6 !\)

Step-by-Step Solution

Answer

The value of 6! is 720.

Step 1: Understanding the Factorial Notation

Factorial notation is represented by the symbol (!). For a positive integer n, the factorial of n (denoted as n!) is the product of all positive integers from 1 up to n.

Step 2: Setting Up the Expression

The given expression is 6! This means it is the factorial of 6, which requires multiplying all positive whole numbers from 1 to 6.

Step 3: Calculating the Factorial

Start multiplying the sequence of numbers from 1 to 6. Begin with 1×2=2, then multiply by 3 to get 6, and continue until the last number. 1. Multiply 1 2. Multiply by 2: 1×2=2 3. Multiply by 3: 2×3=6 4. Multiply by 4: 6×4=24 5. Multiply by 5: 24×5=120 6. Multiply by 6: 120×6=720

Step 4: Conclusion

The result of multiplying all these numbers together is 720, so the value of 6! is 720.

Key Concepts

Permutation

Permutations are arrangements of objects in a specific order. The concept of permutations is closely related to factorial because calculating permutations often involves factorial operations.
For instance, when you are asked to find out the number of ways to arrange 6 different books on a shelf, you are essentially calculating the permutation of 6 objects. This is represented by 6!, which means you multiply all integers from 1 to 6 together.

• Permutations consider the order of arrangement.
• The formula for permutation when arranging all objects is simply the factorial of the number of objects.
• For example, arranging 6 items completely: 6! = 720.

Understanding permutations helps in problems where the sequence or order of items matters.

Combinatorics

Combinatorics is a branch of mathematics dealing with counting, arrangement, and combination of objects. It's a broader concept encompassing permutations and combinations.
While permutations are about arrangement, combinations are concerned with the selection of objects without regard to order.

• Combinatorics often involves calculations with factorials.
• In combinatorics, the factorial function is used to handle calculations involving permutations and combinations.
• For permutations, the order matters, whereas for combinations, it doesn't.

Suitable for solving problems in probability, counting principles, and even understanding the basic math of complex systems.

Mathematics education

Mathematics education focuses on teaching and learning mathematics effectively. Understanding the importance of concepts like factorial, permutation, and combinatorics is part of a solid math foundation.
In education,

• Emphasis is placed on building a strong conceptual understanding.
• Students learn not just how to compute, but also to understand why methods work.
• Practical applications and problem-solving enhance learning processes.

By engaging with problems involving factorials, students can develop valuable skills for higher mathematics, including logical thinking and analytical skills.