Question

Suppose that \(m\) and \(n\) are integers, with \(0 \leq m \leq n .\) The binomial coefficient \(\left(\begin{array}{l}n \\ m\end{array}\right)\) is the coefficient of \(t^{m}\) in the expansion of \((1+t)^{n} ;\) that is, $$ (1+t)^{n}=\sum_{m=0}^{n}\left(\begin{array}{l} n \\ m \end{array}\right) t^{m} $$ From this definition it follows immediately that $$ \left(\begin{array}{l} n \\ 0 \end{array}\right)=\left(\begin{array}{l} n \\ n \end{array}\right)=1, \quad n \geq 0 $$ For convenience we define $$ \left(\begin{array}{r} n \\ -1 \end{array}\right)=\left(\begin{array}{c} n \\ n+1 \end{array}\right)=0, \quad n \geq 0 $$

Short answer

#tag_content# The binomial coefficient, \(\left(\begin{array}{l}n\\m\end{array}\right)\), represents the number of ways to choose \(m\) items from a set of \(n\) items. It is related to the expansion of \((1+t)^n\) as the coefficient of the \(t^m\) term, resulting in the sum of terms as \((1+t)^n = \sum_{m=0}^{n}\left(\begin{array}{l}n\\m\end{array}\right) t^m\). This expansion helps study various mathematical and combinatorial properties associated with the binomial coefficient.

Step-by-step solution

Understanding the binomial coefficient

The binomial coefficient, \(\left(\begin{array}{l}n\\m\end{array}\right)\), is a way to represent the number of different ways to choose \(m\) items from a set of \(n\) items. It is also denoted as \(_nC_m\) or \(C^n_m\). It is crucial in the study of combinatorics and is used in various branches of mathematics.

Relating the binomial coefficient to the expansion of \((1+t)^n\)

According to the problem statement, the binomial coefficient is the coefficient of \(t^m\) in the expansion of \((1+t)^n\). In other words, the sum of the terms in the expansion of \((1+t)^n\) is given by: $$ (1+t)^n = \sum_{m=0}^{n}\left(\begin{array}{l}n\\m\end{array}\right) t^m = \left(\begin{array}{l}n\\0\end{array}\right) t^0 + \left(\begin{array}{l}n\\1\end{array}\right) t^1 + \cdots + \left(\begin{array}{l}n\\n\end{array}\right) t^n $$

Understanding the properties of the binomial coefficient

From the given information, we can see that \(\left(\begin{array}{l}n\\0\end{array}\right)=\left(\begin{array}{l}n\\n\end{array}\right)=1\) for all non-negative integers \(n\). This makes sense mathematically, as there is only one way to choose 0 items from a set of \(n\) items (i.e., not choosing any item), and there is only one way to choose all \(n\) items from a set of \(n\) items (i.e., choosing all of them).

Defining the binomial coefficient for special cases

For convenience and to maintain consistency with the range of possible values of \(m\), we define \(\left(\begin{array}{r}n\\-1\end{array}\right)=\left(\begin{array}{c}n\\n+1\end{array}\right)=0\) for non-negative integers \(n\). This ensures that the binomial coefficient is well-defined when considering series expansion and combinatorial operations.

Key concepts

Combinatorics and the Binomial Coefficient

Combinatorics is a field of mathematics focused on counting, combination, and permutation of sets. A fundamental concept within combinatorics is the binomial coefficient, which appears in a variety of counting problems. The binomial coefficient, represented as \(\binom{n}{m}\), tells us how many ways we can choose \(m\) items from a larger set of \(n\) distinct items regardless of order.

For instance, if we want to determine how many different committees of 3 members can be formed from a group of 5 people, we would calculate the binomial coefficient \(\binom{5}{3}\). This type of question is common in lottery probabilities, game strategies, and even in constructing statistical models. Combinatorics isn't just about numbers; it's about the arrangement and selection within a structured context—which is exactly what the binomial coefficient helps to quantify.

Polynomial Expansion

Polynomial expansion is a core concept in algebra that involves writing out a polynomial that has been raised to a power in its expanded form. For example, the expansion of \( (1 + t)^n \) is the expression of this binomial to the nth power as a sum of terms. The coefficients of each term in this expansion are precisely the binomial coefficients.

The relation of binomial coefficients to polynomial expansion is particularly seen in the binomial theorem, which gives a formula for the expansion of the expression \( (1 + t)^n \) in powers of \( t \). This brings out a vital understanding of how algebraic expressions grow as they are raised to higher powers and finds practical applications in calculus, especially in finding Taylor series and making approximations for functions.

Mathematical Properties of the Binomial Coefficient

The binomial coefficient, being integral to combinatorics and polynomial expansion, has unique mathematical properties that make it a valuable tool. One of the key properties is symmetry, where \(\binom{n}{m} = \binom{n}{n-m}\), reflecting the fact that choosing \(m\) items from \(n\) is the same as leaving out \(n-m\) items.

Another property is that the sum of any two adjacent binomial coefficients in the same row of Pascal's Triangle is equal to the binomial coefficient directly below them in the next row, expressed as \(\binom{n}{m} + \binom{n}{m+1} = \binom{n+1}{m+1}\). These properties not only provide a deeper understanding of combinatorial logic but also simplify computations within various mathematical contexts.

Series Expansion

Series expansion in mathematics allows us to write complex functions as an infinite sum of terms for easier computation and analysis. A famous example is the expansion of a function into a Taylor or Maclaurin series. Series expansion often uses binomial coefficients, especially when dealing with polynomial functions.

In the context of binomial expansion, the series \(\sum_{m=0}^{n}\binom{n}{m} t^m\) is a finite series comprising terms up to \(n\)th power. This is significant in calculus for approximating complex functions around a point and in engineering for signal processing. Understanding the role of the binomial coefficient in series expansion is crucial for studying phenomena that can be modeled by polynomials or approximated by a series, such as physics simulations and economic forecasting models.