Problem 7 - College Algebra with Corequisite Support

Question

For the following exercises, evaluate the binomial coefficient. \(\left(\begin{array}{l}7 \\ 4\end{array}\right)\)

Step-by-Step Solution

Answer

The binomial coefficient \( \binom{7}{4} \) is 35.

Step 1: Understand the binomial coefficient notation

The binomial coefficient \( \binom{n}{k} \) is a mathematical term that represents the number of ways to choose \( k \) elements from a set of \( n \) elements without regard to the order of selection.

Step 2: Use the binomial coefficient formula

The formula to calculate a binomial coefficient is \( \binom{n}{k} = \frac{n!}{k!(n-k)!} \). For our exercise, we have \( n = 7 \) and \( k = 4 \). Substitute these values into the formula: \( \binom{7}{4} = \frac{7!}{4!(7-4)!} \).

Step 3: Calculate the factorials

Calculate the factorials needed for the formula. - \( 7! = 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1 = 5040 \)- \( 4! = 4 \times 3 \times 2 \times 1 = 24 \)- \( (7-4)! = 3! = 3 \times 2 \times 1 = 6 \)

Step 4: Substitute the factorials into the formula

Substitute the calculated factorials back into the formula: \[ \binom{7}{4} = \frac{5040}{24 \times 6} \]

Step 5: Simplify the expression

Simplify the fraction: - First calculate the denominator: \( 24 \times 6 = 144 \).- Then calculate the division: \( \frac{5040}{144} = 35 \).

Step 6: Conclude the solution

The value of the binomial coefficient \( \binom{7}{4} \) is 35.

Key Concepts

Factorials

Factorials are an integral part of mathematics, frequently used in calculations involving permutations and combinations. When we talk about factorials, we represent them using the symbol "!". For an integer \( n \), its factorial, denoted as \( n! \), is the product of all positive integers less than or equal to \( n \). For example, \( 4! = 4 \times 3 \times 2 \times 1 = 24 \). This means that we multiply all numbers starting from 4 down to 1.
Factorials are especially useful in calculating the number of arrangements of a set of items. Since they can become quite large very quickly, they are a powerful tool for understanding concepts in probability and calculus. Here are some key properties of factorials:

• 0! is defined as 1, by convention.
• The factorial operation is only defined for non-negative integers.

Although factorials themselves are simple to compute, they lay the groundwork for more complex mathematical topics, such as permutations and binomial coefficients.

Combinatorics

Combinatorics is the vast field of mathematics dealing with counting, arrangement, and combination of objects. It helps answer questions related to how many ways we can arrange a set of items, or how many possible selections can be made from a larger group. In our exercise, combinatorics is used to evaluate the binomial coefficient, which tells us how many ways we can choose a subset of items from a larger set.
One of the basic tools used in combinatorics is the concept of combinations, denoted as \( \binom{n}{k} \), which represents the number of ways to choose \( k \) items from \( n \) items without regard to order. This is crucial in understanding real-world problems where order does not matter but selection does.

• The order of selection is not important in combinations.
• Combinatorics is essential in probability, algorithm design, and optimization problems.

Combinatorics blends intuition with mathematical rigor, making it essential in various fields, from computer science to engineering.

Binomial Theorem

The binomial theorem provides a formula for expanding expressions raised to a power, specifically for binomials, which are expressions with two terms. The theorem states that for any positive integer \( n \),\[(x + y)^n = \sum_{k=0}^{n} \binom{n}{k} x^{n-k} y^k.\]This expansion involves binomial coefficients, represented as \( \binom{n}{k} \), showing up as the weights or coefficients in the expression.
The binomial theorem is not just theoretical, but highly applicable:

• It allows simplification of polynomial expressions.
• Aids in probability calculations involving multiple trials.
• Helps in understanding distribution patterns in statistics.

Understanding the binomial theorem equips students with the ability to tackle complex algebraic expressions and analyze probabilistic events with ease. Using it unlocks a whole new way to decipher patterns and predictions in mathematical studies.