Problem 2 - A Graphical Approach to Precalculus with Limits

Question

Evaluate the following. In Exercises 17 and 18 , express the answer in terms of n. Do not use a calculator. $$\frac{5 !}{2 ! 3 !}$$

Step-by-Step Solution

Answer

The evaluated value is 10.

Step 1: Understand Factorial Notation

Factorial notation "!" implies the product of all positive integers up to a number. For example, \( n! = n \times (n-1) \times (n-2) \times \cdots \times 2 \times 1 \).

Step 2: Write Out the Factorials

Write \(5!\) as \( 5 \times 4 \times 3 \times 2 \times 1 \), \(2!\) as \( 2 \times 1 \), and \(3!\) as \( 3 \times 2 \times 1 \).

Step 3: Substitute Factorials into the Expression

Substitute the expanded forms of the factorials into the expression: \( \frac{5!}{2! \times 3!} = \frac{5 \times 4 \times 3 \times 2 \times 1}{(2 \times 1)(3 \times 2 \times 1)} \).

Step 4: Simplify the Expression

Cancel the common terms from the numerator and the denominator. The \(3 \times 2 \times 1\) cancels with the corresponding terms in the denominator, leaving \( \frac{5 \times 4}{2 \times 1} \).

Step 5: Evaluate the Simplified Expression

Calculate \( \frac{5 \times 4}{2 \times 1} \) which equals \( \frac{20}{2} = 10 \).

Key Concepts

Simplification Techniques

Simplification techniques are essential in mathematics for reducing complex expressions to their simplest form. When dealing with factorials in combinatorial expressions like \( \frac{5!}{2!3!} \), it is important to understand how to cancel terms. Start by writing out all the factorials involved. Since \( 5! = 5 \times 4 \times 3 \times 2 \times 1 \), \( 2! = 2 \times 1 \), and \( 3! = 3 \times 2 \times 1 \), substitute these into the original expression. At this point, we have \( \frac{5 \times 4 \times 3 \times 2 \times 1}{(2 \times 1)(3 \times 2 \times 1)} \).

• Identify terms that appear in both the numerator and the denominator.
• Cancel these common terms to reduce the expression. This often involves dividing the numerator and the denominator by these common values.
• In this case, most of \( 3 \times 2 \times 1 \) is available both above and below the division line, letting them cancel out directly.

After cancellation, you're left with a much simpler expression: \( \frac{5 \times 4}{2 \times 1} \). From here, directly calculate using basic arithmetic operations.

Factorials in Combinatorics

Factorials in combinatorics serve as the backbone for calculating permutations and combinations, which are methodical ways of arranging and selecting items. These calculations often involve simplifying a factorial expression. Consider the expression \( \frac{5!}{2!3!} \) as an example: it's a typical form found in binomial coefficients, known as \( \binom{n}{k} \), where you choose \( k \) items from a set of \( n \) items.

• To understand factorial usage, note: \( n! \) can be used to determine the number of ways to arrange \( n \) unique items.
• In a combination or permutation setting, you often break this down into overlapping factorials to remove redundant arrangements.

This is why you see expressions like \( \frac{n!}{k!(n-k)!} \) in contexts such as choosing subsets. Applying similar reasoning, \( \frac{5!}{2!3!} \) ends up representing something akin to partitioning 5 objects into groups of 2 and 3.

Mathematical Expressions

Mathematical expressions often involve factorials, especially in combinatorial problems or calculus. A mathematical expression like \( \frac{5!}{2!3!} \) needs to be interpreted using the operations implied by the factorial and division symbols. When working with such expressions:

• Recognize the structure: Factorials lead to large numbers, but simplifying can help manage this.
• Use simplification techniques until you reach a solvable expression.
• Factorials are key in simplifying these expressions, allowing you to cancel and reduce.

By consistently practicing rewriting and simplifying factorial expressions, you can make sense of what these expressions mean and compute them effectively. This mathematical fluency makes handling complex problems more intuitive. Always remember that simplification makes calculations manageable.