Problem 64 - College Algebra with Modeling and Visualization

Question

Evaluate the expression. \(\left(\begin{array}{l}9 \\ 4\end{array}\right)\)

Step-by-Step Solution

Answer

The expression evaluates to 126.

Step 1: Understand the Binomial Coefficient

The expression \( \binom{n}{k} \) represents the number of combinations of \( n \) items taken \( k \) at a time. It is called a binomial coefficient and can be calculated using the formula \( \binom{n}{k} = \frac{n!}{k!(n-k)!} \).

Step 2: Identify the Values

In the given expression \( \left(\begin{array}{l}9 \ 4\end{array}\right) \), we have \( n = 9 \) and \( k = 4 \).

Step 3: Calculate the Factorials

Calculate \( 9! \), \( 4! \), and \( (9-4)! = 5! \).- \( 9! = 9 \times 8 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1 = 362880 \).- \( 4! = 4 \times 3 \times 2 \times 1 = 24 \).- \( 5! = 5 \times 4 \times 3 \times 2 \times 1 = 120 \).

Step 4: Substitute Factorials into the Formula

Substitute the calculated factorials into the binomial coefficient formula:\[ \binom{9}{4} = \frac{9!}{4!5!} = \frac{362880}{24 \times 120} \].

Step 5: Simplify the Expression

Perform the multiplication in the denominator: \( 24 \times 120 = 2880 \).Now, divide the numerator by the denominator: \( \frac{362880}{2880} = 126 \).

Step 6: Conclusion

The expression evaluates to 126. Thus, \( \binom{9}{4} = 126 \).

Key Concepts

Combinatorics

Combinatorics is a fascinating area of mathematics that deals with counting, arranging, and combination of objects. It is widely used in various fields, such as computer science, engineering, and even recreational puzzles. In this context, the binomial coefficient, represented as \( \binom{n}{k} \), is a core concept.

• Definition: The binomial coefficient \( \binom{n}{k} \) represents the number of ways to choose \( k \) elements from a set of \( n \) distinct elements.
• Application: It is crucial in combinatorial problems where the order does not matter, unlike permutations where the order is important.
• Formula: The binomial coefficient can be calculated using the formula \( \binom{n}{k} = \frac{n!}{k!(n-k)!} \).

Understanding combinatorics helps solve problems that involve combinations and arrangements efficiently, providing essential skills for various logical and analytical tasks.

Factorials

Factorials are an essential concept in combinatorics and are a mathematical operation used to represent the product of all positive integers up to a given number \( n \). The factorial of a number \( n \) is denoted as \( n! \).

• For any positive integer \( n \), \( n! = n \times (n-1) \times (n-2) \times \cdots \times 1 \).
• The factorial operation is pivotal when calculating the binomial coefficient, as it allows the determination of possible arrangements.
• Special Case: The factorial of zero is defined as \( 0! = 1 \), providing a foundation for combinatorial mathematics even when selecting zero items.

By understanding factorials, students can efficiently determine the number of permutations and combinations for various mathematical problems.

Mathematics Education

Mathematics education aims to build a deep understanding of various concepts, including combinatorics and factorials, emphasizing their practical applications in problem-solving and real-life situations.

• Conceptual Understanding: It's essential for students to not only memorize formulas but to understand why and how they work.
• Practical Application: Applying these concepts to real-world scenarios enhances learning and retention.
• Problem Solving: Developing problem-solving skills through exercises like calculating binomial coefficients prepares students for complex analytical challenges in the future.

Incorporating interactive and engaging methods in mathematics education fosters an environment where students can appreciate the beauty of mathematical logic and reasoning, making learning more effective and enjoyable.