Problem 7 - Algebra 2 and Trigonometry

Question

In \(3-22,\) evaluate each expression. $$ \frac{10 !}{3 !} $$

Step-by-Step Solution

Answer

The expression evaluates to 604800.

Step 1: Understand Factorials

A factorial, denoted by an exclamation mark (!), is the product of all positive integers up to a specified number. For example, \(n! = n \times (n-1) \times (n-2) \times \ldots \times 1\). So, we have to calculate \(10!\) and \(3!\).

Step 2: Calculate 10!

\(10!\) is the factorial of 10, which means multiplying all integers from 1 to 10: \(10 \times 9 \times 8 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1\). Calculating gives us: \[ 10! = 3628800 \]

Step 3: Calculate 3!

\(3!\) is the factorial of 3, which means multiplying all integers from 1 to 3: \(3 \times 2 \times 1\). Calculating gives us: \[ 3! = 6 \]

Step 4: Divide 10! by 3!

Now, take the calculated factorials and divide \(10!\) by \(3!\): \[ \frac{10!}{3!} = \frac{3628800}{6} \]Calculating the division results in 604800.

Key Concepts

Factorial Division

Factorial division involves dividing one factorial by another. A factorial is the result of multiplying a series of descending natural numbers up to a certain number, denoted with an exclamation mark. For instance, \(n!\) implies multiplying all whole numbers up to \(n\), like \(n \times (n-1) \times ... \times 1\). To compute \(\frac{10!}{3!}\), you first calculate each factorial separately. Then, divide the results. In this instance, \(10!\) equates to 3628800 and \(3!\) to 6. By dividing these, you find: \(\frac{3628800}{6} = 604800\). This process is valid in simplifying expressions or solving problems involving permutations and combinations, emphasizing the practical utility of factorial division in mathematics.

Integer Multiplication

When dealing with factorials, integer multiplication is a fundamental operation. In essence, factorial calculation breaks down into multiplying integers in a series. Multiplying integers is straightforward as it involves repeated addition of the number as per the multiplier. Consider \(10!\) where each integer from 1 to 10 is multiplied in a chain. It begins at 1 and sequentially accumulates each succeeding integer's product, like this:

• \(1 \times 2 = 2\)
• \(2 \times 3 = 6\)
• ... continuing until \(10\) is reached

Each step extends the multiplication, culminating in a large factorial product. Ensuring every step follows order and includes all integers down to one is crucial, especially in significantly larger factorials, accentuating the importance of accurate multiplication for precise outcomes.

Math Calculations

Math calculations often involve multiple steps when dealing with factorial division. The calculation sequence generally includes computing individual factorials followed by division. For \(\frac{10!}{3!}\), begin by evaluating these factorials separately. Start with ordinary multiplication to find \(10!\), which is quite immense, at 3628800. Then calculate \(3!\), obtaining a smaller but essential number, 6. The division \(\frac{3628800}{6} = 604800\) finalizes the calculation. This orderly approach simplifies complex expressions while enhancing understanding of the underlying principles. Comprehensive calculations also highlight the elegance and utility of mathematics in different practical and theoretical contexts, as solutions to permutations or probability problems.