Problem 18 - Beginning and Intermediate Algebra

Question

Evaluate each binomial coefficient. $$\left(\begin{array}{l}8 \\\5\end{array}\right)$$

Step-by-Step Solution

Answer

The binomial coefficient \(\binom{8}{5}\) can be calculated using the formula \(\binom{n}{k} = \frac{n!}{k!(n-k)!}\). By plugging in the values \(n = 8\) and \(k = 5\), we calculate the factorials and simplify the equation: \(\binom{8}{5} = \frac{8!}{5!(8-5)!} = \frac{40320}{120 \times 6} = 56\). Therefore, the value of the given binomial coefficient is \(56\).

Step 1: Identify n and k values

In the given binomial coefficient formula, we are given \(n = 8\) and \(k = 5\).

Step 2: Calculate Factorials

To evaluate the binomial coefficient, first, we need to calculate the factorials of \(n\), \(k\), and \((n-k)\). The factorial of a number is the product of all positive integers less than or equal to that number. \(8! = 8 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1 = 40320\) \(5! = 5 \times 4 \times 3 \times 2 \times 1 = 120\) \((8-5)! = 3! = 3 \times 2 \times 1 = 6\)

Step 3: Plug in values into the binomial coefficient formula

Now that we have calculated the factorials, we can plug in these values into the binomial coefficient formula: \[\binom{8}{5} = \frac{8!}{5!(8-5)!} = \frac{40320}{120 \times 6}\]

Step 4: Simplify the equation

Lastly, we will simplify the equation: \[\binom{8}{5} = \frac{40320}{720} = 56\] So, the value of the given binomial coefficient is 56.

Key Concepts

Factorials

Factorials are essential to calculating binomial coefficients. They are represented by the symbol ``!``. A factorial of a non-negative integer \( n \) is the product of all positive integers less than or equal to \( n \). For example, the factorial of \( n=5 \), expressed as \( 5! \), is calculated as:

• \( 5! = 5 \times 4 \times 3 \times 2 \times 1 = 120 \)

Factorials grow quickly. For instance, \( 8! \) is:

• \( 8! = 8 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1 = 40320 \)

Factorials are critical in expressions that involve arrangements and combinations, making them the backbone of many combinatorial concepts.

Combinatorics

Combinatorics is the branch of mathematics dealing with counting, arranging, and finding patterns in sets. It's all about combinations and permutations. When we use the term combinations, it refers to how we can select items from a group where order doesn’t matter.
This contrasts with permutations where the order does matter. For example, if you have a set of three letters \( \{A, B, C\} \), combinations could be choosing any two letters, like \( \{A, B\} \), \( \{B, C\} \), or \( \{A, C\} \).
Combinatorics helps us to understand "how many?" for different arrangement scenarios, making it indispensable in probability and various practical applications like cryptography and algorithm design.

Binomial Theorem

The binomial theorem provides a powerful way to expand expressions of the form \((x+y)^n\).It tells us that such expressions can be expanded into a sum involving terms of the form:

• \[ \binom{n}{k} x^{n-k}y^k \]

Each term in the expansion consists of a binomial coefficient \(\binom{n}{k}\), which is calculated using factorials.

• \(\binom{n}{k} = \frac{n!}{k!(n-k)!}\)

For example, expanding \((x+y)^2\) leads to:

• \((x+y)^2 = x^2 + 2xy + y^2\)

The coefficients (1, 2, 1) are binomial coefficients. The binomial theorem is not only a mathematical curiosity but also a vital tool in fields such as statistics, finance, and any area where predictions or analyses can be modeled by polynomial expansions.