Problem 7 - College Algebra

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: Understanding the Problem

We need to evaluate the binomial coefficient \( \binom{7}{4} \). The binomial coefficient represents the number of ways to choose 4 elements from a set of 7 elements, regardless of order.

Step 2: Using the Binomial Coefficient Formula

The binomial coefficient \( \binom{n}{k} \) is calculated using the formula:\[\binom{n}{k} = \frac{n!}{k! (n-k)!}\]where \( n! \) denotes the factorial of \( n \). In this problem, \( n = 7 \) and \( k = 4 \).

Step 3: Calculating Factorials

First, we calculate the required factorials:\[ 7! = 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1 = 5040 \] \[ 4! = 4 \times 3 \times 2 \times 1 = 24 \]\[ (7-4)! = 3! = 3 \times 2 \times 1 = 6 \]

Step 4: Plugging Values into the Formula

Substitute the factorial values into the binomial coefficient formula:\[\binom{7}{4} = \frac{7!}{4! \times 3!} = \frac{5040}{24 \times 6}\]

Step 5: Simplifying the Expression

Calculate the denominator:\[ 24 \times 6 = 144 \]Now, substitute back:\[\binom{7}{4} = \frac{5040}{144}\]Divide to find the result:\[ \frac{5040}{144} = 35 \]

Step 6: Conclusion

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

Key Concepts

Factorial

In mathematics, the term 'factorial' is symbolized by an exclamation mark (!). It represents the product of all positive integers up to a specified number, for example, the factorial of 3, written as 3!, is equal to 3×2×1. This equals 6. In our given exercise, calculating factorials is an essential step to find the binomial coefficient.
Factorials are used widely in statistics and calculus because they help us understand how many ways we can arrange or order a particular set of items. You can think of a factorial as the number of ways to arrange n objects in different sequences.

• Example of usage: Calculating the number of permutations.
• Factorials grow very rapidly, e.g., 5! = 120 and 7! = 5040.
• Special case: 0! is defined to be 1, for consistency in equations.

Factorials play a pivotal role when working with binomial coefficients, as we break down a complex question into manageable steps, ultimately leading to a simplified solution.

Combinatorics

Combinatorics is the mathematical study of counting, arranging, and grouping objects. It's a fundamental concept that is at the heart of the binomial coefficient problem.
In our exercise, combinatorics comes into play as we try to determine how many ways we can choose a subset of 4 items from a group of 7 items. This problem of selection out of a larger group without regard to the arrangement is a classic example of a combinatorial problem.

• Combination: Choosing items where the order doesn't matter.
• Permutation: An arrangement of objects where order does matter.
• The binomial coefficient symbol, \( \binom{n}{k} \), represents combinations.

Understanding combinatorics gives us the tools to solve complex problems involving probability, optimization, and statistics. It provides a 'step-by-step' blueprint to unlock the solutions to many real-world challenges.

Permutations

Permutations refer to the different ways in which a set of objects can be arranged in order. It differs from combinations in that permutations consider the order of objects to be important.
For example, when given the letters A, B, and C, the different permutations include ABC, ACB, BAC, BCA, CAB, and CBA. Thus there are 6 specific arrangements.
Permutations are calculated using factorials, hence the formula for finding permutations of n objects is:\[ P(n) = n! \]When discussing permutation-like situations in our given problem of binomial coefficients, understanding this concept can deeply impact the solution.

• Formula: The permutation formula is \( P(n, k) = \frac{n!}{(n-k)!} \), where you choose k items from n options.
• Used in sequences where the order is a key factor.
• Examples include arranging people in line or seating arrangements.

Delving into permutations helps clarify how order impacts the arrangement possibilities, offering essential insight when differentiating between permutation and combination applications in real problems.