Problem 22 - Precalculus Mathematics for Calculus

Question

Evaluate the expression. $$\left(\begin{array}{l}5 \\\2\end{array}\right)\left(\begin{array}{l}5 \\\3\end{array}\right)$$

Step-by-Step Solution

Answer

The expression evaluates to 100.

Step 1: Understand the Notation

The notation \( \binom{n}{k} \) represents a binomial coefficient, which is read as "n choose k" and is calculated using the formula \( \binom{n}{k} = \frac{n!}{k! (n-k)!} \). This counts the number of ways to choose \( k \) elements from a set of \( n \) elements.

Step 2: Calculate \( \binom{5}{2} \)

To calculate \( \binom{5}{2} \), use the formula \( \binom{5}{2} = \frac{5!}{2!(5-2)!} = \frac{5 \times 4 \times 3!}{2 \times 1 \times 3!} = \frac{20}{2} = 10 \).

Step 3: Calculate \( \binom{5}{3} \)

Similarly, for \( \binom{5}{3} \), use the formula \( \binom{5}{3} = \frac{5!}{3!(5-3)!} = \frac{5 \times 4 \times 3!}{3 \times 2 \times 1 \times 2!} = \frac{20}{6} = 10 \).

Step 4: Multiply the Results

Now that you have both coefficients, multiply them together: \(10 \times 10 = 100\).

Key Concepts

Combinatorics

Combinatorics is the branch of mathematics that deals with counting, arrangement, and combination of objects. It plays a fundamental role in solving problems where we need to calculate the number of ways to perform a particular set of arrangements or selections. In the context of the given problem, combinatorics helps us determine how many different ways we can choose a subset of items from a larger set. For example, if you want to know how many ways you can choose 2 objects out of a pool of 5, combinatorics provides the tools using concepts like the binomial coefficient, which we'll delve deeper into later.

• Counting Principles: This includes basic counting principles such as the rule of product, where the total number of outcomes for multiple independent events can be found by multiplying the number of outcomes for each event.
• Pigeonhole Principle: A useful concept that involves placing items into containers, representing how a specific distribution would lead to at least one container holding more than a certain number of items.

Understanding these concepts can make it easier to grasp how calculations involving arrangements and selections work, especially when dealing with more complex problems.

Factorial Function

The factorial function is a crucial part of understanding combinations, permutations, and other combinatorial objects. Represented as "!", the factorial of a number is the product of all positive integers less than or equal to that number. For instance, 5 factorial (\(5!\)) is calculated as 5 x 4 x 3 x 2 x 1, which equals 120. This function is essential for calculating binomial coefficients because it allows for the enumeration of all possible orders or arrangements of a set.

• Factorial Growth: As numbers increase, their factorials grow extremely fast, which shows their vast utility in combinatorics and probability.
• Simplification: Often in combinatorial problems, the factorial function helps in simplifying expressions, such as in the given problem where we cancel out terms in the numerator and denominator to find the binomial coefficient easily.

Remember that factorials are a building block of many formulas in mathematics, especially those related to counting and arranging objects in order.

Binomial Theorem

The binomial theorem is a powerful tool in algebra and combinatorics, describing the algebraic expansion of powers of a binomial. This theorem is expressed as follows: \[(x + y)^n = \sum_{k=0}^{n} \binom{n}{k} x^{n-k} y^k\]where \( \binom{n}{k} \) is the binomial coefficient. This formula explains how to expand expressions raised to a power in terms of its individual terms. In simpler terms, it allows us to break down exponential expressions into manageable parts by providing the coefficients of each term.

• Application: This theorem finds widespread application in algebra, probability, and even calculus and discrete mathematics.
• Binomial Coefficients: The central role in this theorem is played by the binomial coefficients, such as \(\binom{5}{2}\) and \(\binom{5}{3}\) in the problem, which is a count of the ways to choose elements from a set, a critical aspect of combinatorial calculations.

Understanding how to apply the binomial theorem can simplify complex algebraic problems by relating them back to fundamental combinatorial ideas.