Problem 24 - Algebra and Trigonometry

Question

evaluate each factorial expression. $$ \frac{18 !}{16 !} $$

Step-by-Step Solution

Answer

The value of the expression \(\frac{18!}{16!}\) is 306.

Step 1: Understand the Properties of Factorials

The factorial of a number 'n', denoted as 'n!', is the product of all positive integers from 1 to n. It is important to note that 'n!' is equal to 'n* (n-1)!'. So, we can simplify \(\frac{18!}{16!}\) as this: \(18 * 17 * 16!\) over \(16!\). This cancels out 16!.

Step 2: Simplify the Factorial Expression

Using the property from Step 1, we can simplify \(\frac{18!}{16!}\) to \(18 * 17\). The 16! in the numerator and the denominator cancel each other out. This simplification significantly reduces the complexity of the expression.

Step 3: Calculate the Product

We calculate the product of 18 and 17 which is 306.

Key Concepts

Factorial Properties

Factorials are an interesting and useful concept in mathematics. The factorial of a number, denoted as \( n! \), represents the product of all positive integers from 1 to that number \( n \). For example, \( 5! = 5 \times 4 \times 3 \times 2 \times 1 = 120 \). This principle is used often in permutations, combinations, and various areas of probability and statistics. A crucial property of factorials is that \( n! = n \times (n-1)! \). This means that any factorial can be broken down into smaller factorials, which is very handy for simplifying expressions involving factorials. Understanding this foundational property aids significantly in reducing complexity in various mathematical operations.

Simplifying Expressions

Simplifying expressions is a key skill that helps manage complex calculations efficiently. In the context of factorial expressions like \( \frac{18!}{16!} \), one can leverage factorial properties to simplify. Here, we use the property \( n! = n \times (n-1)! \) to express \( 18! \) as \( 18 \times 17 \times 16! \). This allows us to cancel out the \( 16! \) in the numerator with the \( 16! \) in the denominator of the expression, drastically simplifying it to \( 18 \times 17 \). Simplification reduces computational cost and improves clarity, making it easier to handle more complex problems.

Product Calculation

Once we have simplified an expression, the next step is to calculate the product of the remaining terms. After simplifying \( \frac{18!}{16!} \) to \( 18 \times 17 \), we must calculate this product to get our final answer. Calculating this is straightforward:\[ 18 \times 17 = 306 \] This step completes the problem by reducing a complex fraction to a simple arithmetic multiplication question. Understanding how to perform these calculations and navigate through these operations are fundamental mathematical skills, reinforcing the practical application of algebraic manipulation and arithmetic.