Problem 65 - Beginning and Intermediate Algebra

Question

Find the indicated term of each binomial expansion. Show that \(\left(\begin{array}{l}n \\ n\end{array}\right)=1\) for any positive integer \(n\)

Step-by-Step Solution

Answer

We used the binomial coefficient formula \(\left(\begin{array}{l}n \\\ k\end{array}\right) = \frac{n!}{k!(n-k)!}\) and substituted \(k=n\), giving us \(\left(\begin{array}{l}n \\\ n\end{array}\right) = \frac{n!}{n!(0)!}\). Since \(0!=1\), the expression simplifies to \(\left(\begin{array}{l}n \\\ n\end{array}\right) = \frac{n!}{n!}\), which equals to 1. Hence, \(\left(\begin{array}{l}n \\\ n\end{array}\right)=1\) for any positive integer \(n\).

Step 1: Binomial Coefficient Formula

The binomial coefficient can be calculated using the following combinatorics formula: \[\left(\begin{array}{l}n \\\ k\end{array}\right) = \frac{n!}{k!(n-k)!},\] where \(n!\) denotes the factorial of \(n\) and is calculated as the product of all positive integers up to \(n\). In our case, we need to find the coefficient for \(k = n\), so the expression will be: \[\left(\begin{array}{l}n \\\ n\end{array}\right) = \frac{n!}{n!(n-n)!}.\]

Step 2: Substituting k=n

We will now substitute \(k=n\) in our formula and simplify it: \[\left(\begin{array}{l}n \\\ n\end{array}\right) = \frac{n!}{n!(0)!}.\]

Step 3: Simplifying the expression

As \(0! = 1\) in combinatorics, we can simplify the expression further: \[\left(\begin{array}{l}n \\\ n\end{array}\right) = \frac{n!}{n! \times 1}.\] Now, when dividing \(n!\) by itself, we simply get 1: \[\left(\begin{array}{l}n \\\ n\end{array}\right) = \frac{n!}{n!} = 1.\]

Step 4: Conclusion

We have shown that the binomial coefficient \(\left(\begin{array}{l}n \\\ n\end{array}\right)\) is equal to 1 for any positive integer \(n\).

Key Concepts

Combinatorics

Combinatorics is the branch of mathematics that deals with counting, arranging, and combination of objects. It comes in handy when analyzing possibilities in various scenarios. A fundamental part of combinatorics is the concept of a binomial coefficient, often used to calculate the number of ways to choose a set of items from a larger set, regardless of order. This calculation is central to many problems regarding permutations and combinations. By leveraging combinatorics, we can efficiently solve problems involving large datasets or complex structures by breaking them down into manageable parts.

Factorial

The factorial, denoted as \(n!\), represents the product of all positive integers up to a given number \(n\). For instance, \(5! = 5 \times 4 \times 3 \times 2 \times 1 = 120\). It's important in various calculations in mathematics, especially in combinatorics.

• If \(n = 0\), then \(0!\) is defined as 1.

The factorial grows rapidly with an increase in \(n\), which makes it an interesting function to study in mathematical analysis. Factorials are used extensively in the calculation of permutations and combinations, providing the foundation for the computation of binomial coefficients.

Binomial Expansion

Binomial expansion refers to expanding powers of a binomial expression. A binomial expression is simply two terms added together, like \((a+b)\). According to the Binomial Theorem, any power \(n\) of a binomial \((a+b)^n\) can be expanded into a sum involving binomial coefficients:\[(a+b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k\]Here, \(\binom{n}{k}\) is the binomial coefficient, representing the number of ways the term can be constructed. Binomial expansion is vital in algebra and calculus since it simplifies the multiplication of polynomial expressions and the evaluation of limits, integrals, and other advanced topics.

Positive Integers

Positive integers are the numbers starting from 1 and continuing upwards, such as 1, 2, 3, and so on. They do not include zero, negative numbers, or fractions - only whole numbers greater than zero.

• They form the basis of many mathematical concepts and operations.
• In the context of combinatorics, positive integers often denote the number of items in a set or the specific term being considered.

Understanding positive integers is crucial when delving into various mathematical theories and functions since they establish the foundations of counting and measuring in math.