Problem 14 - Algebra and Trigonometry

Question

Evaluate each factorial expression. \(\frac{5 ! 8 !}{3 !}\)

Step-by-Step Solution

Answer

806400

Step 1 - Understand Factorials

Factorials are the product of an integer and all the integers below it. For example, 5! (5 factorial) is equal to 5 × 4 × 3 × 2 × 1.

Step 2 - Calculate Individual Factorials

Calculate each factorial separately: \(5! = 5 \times 4 \times 3 \times 2 \times 1 = 120\) \(8! = 8 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1 = 40320\) \(3! = 3 \times 2 \times 1 = 6\)

Step 3 - Substitute Factorials into Expression

Substitute the calculated factorials back into the original expression: \( \frac{5! \times 8!}{3!} = \frac{120 \times 40320}{6} \)

Step 4 - Simplify the Expression

First, multiply the numerators: \(120 \times 40320 = 4838400\)Next, divide by the denominator: \( \frac{4838400}{6} = 806400 \)

Key Concepts

Factorial Calculation

The concept of factorials is essential in many areas of mathematics and is often denoted using the exclamation mark (!). A factorial of a non-negative integer is found by multiplying that integer by all positive integers less than it. For example, to calculate the factorial of 5 (denoted as 5!), you multiply 5 × 4 × 3 × 2 × 1, which equals 120.
Here are some steps:

• Identify the integer (e.g., 5 for 5!)
• Multiply it by every positive integer below it (e.g., 4, 3, 2, 1 for 5!)
• Continue until you reach 1

So, 8! is calculated as 8 × 7 × 6 × 5 × 4 × 3 × 2 × 1, resulting in 40320. The factorial of 3 (3!) is simply 3 × 2 × 1 = 6. Remember, calculating factorials can be intensive, but breaking it down step-by-step makes it manageable.

Simplifying Expressions

When faced with a complicated expression involving factorials, the goal is to simplify it step-by-step. Once you've calculated the individual factorials, the next step is substitution. Replace the factorials in the expression with their numerical values.
In our example, \(\frac{5! \times 8!}{3!}\) becomes \(\frac{120 \times 40320}{6}\).
The simplification process involves:

• Calculating each factorial value independently
• Substituting these values back into the original expression

This makes it easier to handle, as dealing with large products can be overwhelming if not broken down. Next, simplify the expression by performing the indicated operations in sequence. This involves combining the numerators first and then dividing by the denominator.

Multiplication and Division in Algebra

Handling multiplication and division within algebraic expressions is all about following the order of operations. When you multiply two numbers, you combine their values, and when you divide one number by another, you determine how many times the denominator fits into the numerator.

In the example given, \(\frac{120 \times 40320}{6}\), you first focus on the multiplication:

• Multiply 120 by 40320 to get 4838400

Then, you proceed with the division:

• Divide 4838400 by 6, resulting in 806400.

This way, you effectively break down the problem into smaller, more manageable steps. Always carrying operations in sequence ensures accuracy and clarity, making the arithmetic much simpler and helping to avoid mistakes.