Problem 28 - Beginning and Intermediate Algebra

Question

Evaluate each binomial coefficient. $$\left(\begin{array}{l}7 \\\0\end{array}\right)$$

Step-by-Step Solution

Answer

The value of the binomial coefficient \(\binom{7}{0}\) is 1, which can be found using the formula \(\binom{n}{k} = \frac{n!}{k!(n-k)!}\) and calculating the required factorials.

Step 1: Identify the values of n and k

In this binomial coefficient, our values are \(n = 7\) and \(k = 0\).

Step 2: Calculate the factorials

To calculate the binomial coefficient, we need the factorials of \(n\), \(k\), and \((n-k)\). Calculate these factorials: \(7! = 7 * 6 * 5 * 4 * 3 * 2 * 1 = 5040\) \(0! = 1\) (by definition) \((7 - 0)! = 7! = 5040\) (already calculated above)

Step 3: Apply the binomial coefficient formula

Now that we have calculated the factorials, we can apply the binomial coefficient formula: \[\binom{7}{0} = \frac{7!}{0!(7-0)!} = \frac{5040}{1 * 5040}\]

Step 4: Simplify and find the answer

Simplify the expression in Step 3: \[\binom{7}{0} = \frac{5040}{5040} = 1\] Therefore, the value of the given binomial coefficient is 1.

Key Concepts

Factorial

A factorial, denoted by an exclamation mark (!), is a fundamental concept in mathematics, particularly in combinatorics and probability. It is used when arranging or selecting objects in various ways. The factorial of a non-negative integer, such as "n", is the product of all positive integers less than or equal to "n".
For example, the factorial of 5 (written as 5!) is the product of all integers from 5 down to 1, which equals 120:

• 5! = 5 × 4 × 3 × 2 × 1 = 120

The factorial of 0 is defined as 1, which is a crucial exception to remember. This special case allows various mathematical formulas, including those involving binomial coefficients, to work seamlessly even when zero is involved.
Factorials grow quite rapidly. For this reason, they are often encountered in calculations that relate to counting the number of ways to arrange items or select a subset of items from a larger set.

Binomial Theorem

The binomial theorem is a cornerstone in algebra and combinatorics, providing a way to expand expressions raised to a power. Specifically, it describes the algebraic expansion of powers of a binomial, which is an expression involving two terms, like ewlineewline\((a + b)^n\).Each term in the expansion of a binomial raised to a power involves a binomial coefficient, which is represented by the symbol \( \binom{n}{k} \). These coefficients indicate the number of ways to choose "k" elements from a set of "n" elements, without regard to the order.ewlineewlineThe binomial theorem is expressed as:ewline\((a + b)^n = \sum_{k=0}^{n}\binom{n}{k}a^{n-k}b^k\)ewline Here, \( \sum \) denotes the sum over all terms, with each term containing a unique combination of the powers of "a" and "b". This theorem not only aids in expanding binomials but also reveals deeper insights into the properties of numbers and coefficients that arise in such expansions.

Combinatorics

Combinatorics is the branch of mathematics concerning the study of finite or countable discrete structures. It is used to solve problems related to the arrangement, selection, and operation on sets of elements. At its core, combinatorics is about counting and organizing entities, which is essential in various fields such as computer science, cryptography, and probability theory.One of the most famous tools in combinatorics is the binomial coefficient, represented as \( \binom{n}{k} \). This coefficient calculates the number of ways to choose "k" elements from a set of "n" elements, ignoring the order in which they are selected.Some key points in combinatorics include:

• Permutations: Arranging all elements of a set in all possible ways.
• Combinations: Selecting some elements of a set regardless of order.
• Applications: Used in creating algorithms, optimizing networks, predicting outcomes, and more.

Combinatorics is a foundational element of mathematical study, connecting to many fundamental theories and allowing for the solution of complex logical and quantitative puzzles.