Problem 60 - College Algebra with Modeling and Visualization

Question

Evaluate the expression. \(c(7,5)\)

Step-by-Step Solution

Answer

The value of \(c(7,5)\) is 21.

Step 1: Understand the Symbol

The expression \(c(7,5)\) is a binomial coefficient, which is often written as \(\binom{7}{5}\). It represents the number of ways to choose 5 elements from a total of 7 elements without regard to order.

Step 2: Apply the Binomial Coefficient Formula

The binomial coefficient \(\binom{n}{k}\) is calculated using the formula:\[\binom{n}{k} = \frac{n!}{k!(n-k)!}\]For \(c(7,5)\), we substitute \(n = 7\) and \(k = 5\) into the formula.

Step 3: Substitute the Values

Substitute \(n = 7\) and \(k = 5\) into the formula:\[\binom{7}{5} = \frac{7!}{5! \, (7-5)!}\]This simplifies to:\[\binom{7}{5} = \frac{7!}{5! \, 2!}\]

Step 4: Calculate the Factorials

Calculate the factorials involved:- \(7! = 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1 = 5040\)- \(5! = 5 \times 4 \times 3 \times 2 \times 1 = 120\)- \(2! = 2 \times 1 = 2\)

Step 5: Calculate the Binomial Coefficient

Plug the factorial values back into the binomial coefficient formula:\[\binom{7}{5} = \frac{5040}{120 \times 2}\]Next, calculate \(120 \times 2 = 240\), and then divide \(5040\) by \(240\):\[\binom{7}{5} = \frac{5040}{240} = 21\]

Step 6: Conclusion

The value of the expression \(c(7,5)\) or \(\binom{7}{5}\) is 21. Therefore, there are 21 ways to choose 5 elements from 7 elements.

Key Concepts

Factorials

Factorials are fundamental in mathematics, particularly when dealing with permutations and combinations. A factorial, denoted as \( n! \), is the product of all positive integers up to \( n \). For example, \( 5! \) equals \( 5 \times 4 \times 3 \times 2 \times 1 = 120 \). You can think of a factorial as a process that counts the number of ways to arrange \( n \) distinct objects.

Factorials grow very quickly as the number increases. They are key in calculating binomial coefficients, which help in determining the number of ways to select items from a set.

• \( 0! = 1 \) by definition, which is handy in many calculations.
• \( 1! = 1 \) since there's only one way to arrange a single item.
• Factorials aid in simplifying complex expressions in algebra and calculus.
• Often encountered in the fields of probability and statistics.

Understanding factorials is crucial when exploring the wider applications of combinatorics.

Combinatorics

Combinatorics is a branch of mathematics focused on counting, arrangement, and combination of elements within a set. It plays a critical role in various fields such as computer science, probability, and algorithm design.

Within combinatorics, the concept of choosing elements from a set is essential. This is where binomial coefficients come in, which count how many ways you can choose \( k \) elements from a larger set of \( n \) elements.

There are several basic concepts:

• Permutations: Arrangements of objects where order matters.
• Combinations: Selections of objects where order doesn't matter, using the formula \( \binom{n}{k} = \frac{n!}{k!(n-k)!} \).
• Applications include counting problems and resource allocation scenarios.

Combinatorics provides tools to solve problems by considering all possible arrangements or selections, making it indispensable in both theoretical and applied mathematics.

Binomial Theorem

The Binomial Theorem provides a way to expand expressions that are raised to a power, indicated as \((a + b)^n\). It is closely related to binomial coefficients, as each term in the expansion has a coefficient equal to a binomial coefficient.

The theorem states that: \[(a + b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k\]

Each term in the expansion involves binomial coefficients, which are determined using factorials. This formula is essential for simplifying complicated polynomial expansions and finding specific terms in a binomial expansion without fully expanding it.

Key features of the Binomial Theorem include:

• Provides a systematic approach to expanding binomials.
• Links to Pascal's Triangle, where each row corresponds to the coefficients of an expanded binomial raised to consecutive powers.
• Useful in probability theory, particularly in binomial distributions.
• The coefficients \( \binom{n}{k} \) are also useful in different applications of algebra.

The Binomial Theorem is powerful, allowing not just expansion, but also insights into the structure of polynomial equations.