Problem 4 - College Algebra

Question

In Exercises 1-8, evaluate the given binomial coefficient. $$\left(\begin{array}{l}11 \\ 1\end{array}\right)$$

Step-by-Step Solution

Answer

The binomial coefficient \( \left(\begin{array}{l}11 \ 1\end{array}\right) \) equals 11.

Step 1: Understand the Binomial Coefficient

The binomial coefficient, often pronounced as 'n choose k', gives the number of ways, disregarding order, that \(k\) items can be chosen from \(n\) items. More technically, it calculates the number of unordered outcomes of an experiment (like tossing a coin several times). In this case, the binomial coefficient is written as \( \left(\begin{array}{l}11 \ 1\end{array}\right) \). Here, 11 and 1 are the values of \(n\) and \(k\) respectively.

Step 2: Calculate Factorials

For finding the binomial coefficient, you need to calculate the factorials of \(n\), \(k\) and \(n-k\). Here, \(n!\) = \(11!\), \(k!\) = \(1!\), and \((n-k)!\) = \((11-1)!\) = \(10!\). Factorials are calculated by multiplying all positive integers up to that number. So, \(1!\) = 1, \(10!\) = \(10 × 9 × 8 × 7 × 6 × 5 × 4 × 3 × 2 × 1\), and \(11!\) = \(11 × 10!\).

Step 3: Substitute into the Formula

Substitute values into the formula for binomial coefficient \(\frac{n!}{k!(n-k)!}\). Plug the values, getting \(\frac{11!}{1!(11-1)!}\), which becomes \(\frac{11 × 10!}{1 × 10!}\). The \(10!\) in the numerator and denominator will cancel out, giving 11.

Step 4: Simplify

Simplifying the expression gives the result 11.

Key Concepts

Factorials

Factorials are a foundational concept in mathematics, particularly when working with the binomial coefficient in combinatorics. A factorial of a number, denoted as \( n! \), is the product of all positive integers from 1 up to \( n \). It serves as a way to calculate permutations and combinations effectively.

To understand factorials, let's break it down with simpler numbers. For example:

• \( 3! \) translates to \( 3 \times 2 \times 1 = 6 \)
• \( 4! \) becomes \( 4 \times 3 \times 2 \times 1 = 24 \)

One useful property of factorials is that \( 0! = 1 \), a fact often used in mathematical equations as a base case.

In the context of the binomial coefficient, factorials are vital because they help break down how many permutations or combinations of a certain number of items are possible. By understanding and calculating factorials, students can compute complex mathematical expressions, such as binomial coefficients, with ease.

Combinatorics

Combinatorics is the branch of mathematics focused on counting, arranging, and finding patterns or arrangements. One practical application of combinatorics is determining the number of ways to choose items from a set, which is where the binomial coefficient comes into play. Combinatorics helps solve problems related to probability, graph theory, and coding.

In the case of the binomial coefficient \( \left( \begin{array}{c} n \ k \end{array} \right) \), combinatorics answers the question: "How many ways can we choose \( k \) items from \( n \) items?" This is crucial in fields such as statistics, where understanding possible outcomes is necessary.

By finding combinations, students learn more about:

• Choosing a specific number of items from a larger set
• Understanding the importance of order in arrangements
• Applying these concepts to real-world problems

This knowledge is foundational for many areas in math and science, providing a toolset for tackling quantitative problems.

Permutations

Permutations deal with the arrangement of a set of items where order does matter. Unlike combinations, permutations consider different sequences as distinct arrangements, which increases the possible outcomes. The formula for calculating permutations of \( n \) items taken \( k \) at a time is \( \frac{n!}{(n-k)!} \).

For example, if you have 3 books and want to know the number of ways to arrange all of them on a shelf, permutations come into play:

• Arrange 3 books: \( 3! = 6 \) ways — so you can have different orders like ABC, ACB, BAC, etc.

Permutations are particularly useful when sequences or arrangements matter, such as determining passwords or lock combinations where the order must be precise.

Understanding permutations helps students distinguish between scenarios where order is important and where it is not. Both concepts work hand-in-hand with combinations to give a full picture of arrangements, as evidenced by exercises involving binomial coefficients.