Problem 5 - Algebra 2 and Trigonometry

Question

In \(3-22,\) evaluate each expression. $$ 8 ! \div 3 ! $$

Step-by-Step Solution

Answer

The value of \(8! \div 3!\) is 6,720.

Step 1: Understand the Factorial

The factorial of a number, represented by the exclamation mark (!), is the product of all positive integers up to that number. For example, 5! = 5 × 4 × 3 × 2 × 1.

Step 2: Calculate 8!

To evaluate 8!, multiply all positive integers from 1 to 8. \[8! = 8 imes 7 imes 6 imes 5 imes 4 imes 3 imes 2 imes 1 = 40,320\]

Step 3: Calculate 3!

Next, calculate 3! by multiplying all positive integers from 1 to 3. \[3! = 3 imes 2 imes 1 = 6\]

Step 4: Divide 8! by 3!

Divide the value of 8! by the value of 3! to evaluate the expression. \[\frac{8!}{3!} = \frac{40,320}{6} = 6,720\]

Step 5: Provide the Final Result

The result of the expression \(8! \div 3!\) is 6,720.

Key Concepts

Factorial Division

Factorial division involves dividing the factorial of one number by the factorial of another. This operation is often encountered in algebra problems, particularly in permutations and combinations. To understand factorial division, let's take the concept of dividing two factorial numbers, like we see in the expression \( \frac{8!}{3!} \). Here, we are dividing the product of all numbers up to 8 by the product of all numbers up to 3. The calculation follows these steps:

• First, calculate \(8!\), which is \(8 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1 = 40,320\).
• Then, find \(3!\), which simplifies to \(3 \times 2 \times 1 = 6\).

Finally, divide \(8!\) by \(3!\) using the formula \(\frac{40,320}{6} = 6,720\). The quotient is the end result of the factorial division.

Factorial Calculation

Factorial calculation revolves around finding the product of a sequence of descending natural numbers. It is denoted by the symbol "!". For any positive integer \(n\), the factorial is represented as \(n!\). Here's how you calculate it:

• Identify the number whose factorial you want to calculate, say \(n\).
• Multiply all whole numbers from \( n \) down to 1.

For example, to calculate \(5!\):- Multiply: \(5 \times 4 \times 3 \times 2 \times 1 = 120\).Calculating factorials becomes crucial when solving various algebraic or real-world problems that require the arrangement or selection of objects. Remember, the number zero is a special case in factorials, as 0! is defined to be 1 by convention.

Algebra Problem Solving

Algebra problem solving often requires applying rules and formulas to simplify complex expressions, including those with factorials. By mastering factorial calculations and divisions, one can solve permutation and combination problems efficiently. Approach these algebra problems by:

• Breaking the problem into smaller parts that are easier to handle. Start by calculating any factorials separately.
• Learning how to simplify equations by canceling out terms, similar to reducing fractions in factorial division.

In our example, solving for \(\frac{8!}{3!}\) involves understanding how factorials simplify expressions to make calculations straightforward. You focus on finding integers that can cancel each other to reduce complexity, making it easier to arrive at a solution, thus reinforcing your algebraic skills.