Problem 76 - Algebra 2

Question

Simplify each expression. $$ 4 ! $$

Step-by-Step Solution

Answer

The simplified expression of \(4!\) is 24.

Step 1: Understanding Factorial

In mathematics, the factorial of a positive integer \(n\), denoted by \(n!\), is the product of all positive integers less than or equal to \(n\). That is \(n! = n \times (n-1) \times (n-2) \times \ldots \times 3 \times 2 \times 1\)

Step 2: Applying the Factorial Definition to Our Problem

To find the factorial of 4, or \(4!\), multiply 4 by all the positive integers below it. So, \(4! = 4 \times 3 \times 2 \times 1\)

Step 3: Calculate Result

Now calculate the expression. Starting from left to right, firstly \(4 \times 3 = 12\), then multiply this result by 2, we get \(12 \times 2 = 24\), finally, \(24 \times 1 = 24\) which is our answer.

Key Concepts

Factorial Notation

When you see a number followed by an exclamation mark, like "4!", you're looking at a factorial. Factorial notation is a way to represent the product of all positive integers up to a given number. It's a shorthand used frequently in mathematics, especially in probability and statistics.
\[\]Factorials are simple to understand once you grasp the concept. For example, the expression "4!" stands for \(4 \times 3 \times 2 \times 1\). You take the number and multiply it by each whole number less than itself down to 1.
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• "0!" is a special case and is defined to equal 1.
• This notation helps simplify expressions and calculations.

Recognizing factorial notation is a key step in solving factorial problems.

Integer Multiplication

Multiplication is a foundational operation in mathematics, especially when dealing with factorials. Integer multiplication involves taking two whole numbers and finding their product. When calculating factorials, you'll perform a series of these multiplications.
\[\]Let's consider the factorial of 4, which involves multiplying four integers: 4, 3, 2, and 1. You start with the largest number and multiply it by the next lower number in sequence. For instance, 4 times 3 equals 12.
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• The result (12) is then multiplied by the next integer (2), continuing until you reach 1.
• This technique ensures accurate calculation of the factorial product.

Always verify each step as you progress to avoid mistakes.

Factorial Calculation

Calculating a factorial is straightforward once you understand the process. It involves methodically multiplying a sequence of decreasing positive integers.
\[\]To compute 4!, start by multiplying 4 by 3, yielding 12. Next, take that result and multiply by 2, which gives 24. Finally, multiplying 24 by 1 still results in 24. Therefore, the factorial of 4 is 24.
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• This systematic approach applies to any integer factorial calculation.
• Start from the given number and multiply backwards to 1.

Remember, calculating accurately ensures the correctness of complex mathematical problems.

Mathematical Expressions

Mathematical expressions are combinations of numbers and operations like addition, subtraction, multiplication, and division. In factorial expressions, the main operation is multiplication.
\[\]Mathematical expressions can become more understandable by simplifying them, which is essential for problem-solving. Simplifying a factorial expression means calculating its actual number, like turning "4!" into 24.
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• Simplification helps in interpreting and solving more complex equations.
• A simplified expression is easier to work with in further calculations.

Practicing these simplifications makes you more comfortable handling various mathematical problems.