Problem 51 - Algebra and Trigonometry

Question

Simplify using the Binomial Theorem. Show that \(\left(\begin{array}{l}{n} \\\ {1}\end{array}\right)=\left(\begin{array}{c}{n} \\ {n-1}\end{array}\right)=n\)

Step-by-Step Solution

Answer

Both expressions equal \( n \).

Step 1: Definition of Binomial Coefficients

The binomial coefficient for any two non-negative integers \( n \) and \( k \), is given by \( \binom{n}{k} = \frac{n!}{k!(n-k)!} \).

Step 2: Calculating Binomial Coefficient \( \binom{n}{1} \)

Using the binomial coefficient formula, \( \binom{n}{1} = \frac{n!}{1!(n-1)!} = \frac{n \times (n-1)!}{1 \times (n-1)!} \). The \((n-1)!\) terms cancel each other out, simplifying the expression to \( n \).

Step 3: Calculating Binomial Coefficient \( \binom{n}{n-1} \)

Similarly, \( \binom{n}{n-1} = \frac{n!}{(n-1)!(n-(n-1))!} = \frac{n!}{(n-1)!1!} = \frac{n \times (n-1)!}{(n-1)!} \). Again, the \((n-1)!\) terms cancel each other out, simplifying the expression to \( n \).

Step 4: Conclusion

Both \( \binom{n}{1} \) and \( \binom{n}{n-1} \) simplify to \( n \). Thus, \( \binom{n}{1} = \binom{n}{n-1} = n \), which shows the given equivalence.

Key Concepts

Binomial Coefficient

The binomial coefficient is an essential concept in algebra, especially when using the Binomial Theorem. It represents the number of ways to choose a subset of elements from a larger set, disregarding the order of selection. The notation for the binomial coefficient is \( \binom{n}{k} \), where \( n \) stands for the total number of elements, and \( k \) indicates the size of the subset. This coefficient is calculated using the formula:

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

It's important to realize that in this formula, the exclamation mark denotes a factorial, which we will explore further in the next section. The binomial coefficient is symmetrical, meaning \( \binom{n}{k} = \binom{n}{n-k} \), which is a neat property that can simplify calculations and solutions.

Factorial Notation

Factorial notation is a mathematical operation represented by an exclamation mark. It is used primarily in permutations and combinations. The factorial of a non-negative integer \( n \), denoted as \( n! \), is the product of all positive integers less than or equal to \( n \):

• \( n! = n \times (n-1) \times (n-2) \times \ldots \times 1 \)

The factorial operation helps to determine the number of ways to arrange a set of objects. By convention, \( 0! \) is defined as 1. Factorials grow very quickly with increasing \( n \), which is why they are greatly beneficial for solving problems in combinatorics, such as computing binomial coefficients.
In practical terms, knowing how to work with factorials allows us to simplify expressions like \( \binom{n}{1} \) and \( \binom{n}{n-1} \) to \( n \) by effectively cancelling terms in the numerator and the denominator.

Combinatorics

Combinatorics is the branch of mathematics dealing with combinations of objects. It explores how objects can be arranged or selected under various conditions. Combinatorics includes topics like permutations, combinations, and binomial coefficients.
A fundamental part of combinatorics involves understanding how to count these combinations. With permutations, order matters, while in combinations, it does not. The Binomial Theorem utilizes this concept by offering a way to expand expressions raised to a power, represented usually as \( (x + y)^n \).

• The expansion results in a series termed a binomial expansion.
• The coefficients within this expansion are actually the binomial coefficients \( \binom{n}{k} \).

By unraveling these concepts in combinatorics, students can approach complex counting and probability problems with greater confidence and insight, particularly when working with theorems like the Binomial Theorem.