Binomial theorem (expansion)
The Binomial Theorem, also known as the Binomial Expansion, is a fundamental result in combinatorics and algebra that describes the expansion of a binomial expression raised to a non-negative integer power. The theorem is particularly useful when working with expressions of the form \((a+b)^n \), where \(a\) and \(b\) are real or complex numbers and \(n\) is a non-negative integer.
The Binomial Theorem states that for any non-negative integer \(n\) and any real or complex numbers \(a\) and \(b\),
\( (a + b)^n = \sum_{k=0}^{n} \binom{n}{k} b^k a^{n-k} \)
Here, \( \binom{n}{k} \), read as "n choose k," is a binomial coefficient, which can be computed using the formula:
\( \binom{n}{k} = \frac{n!}{K!(n-k)!} \)
In this formula, \(n!\) denotes the factorial of \(n\), which is the product of all positive integers up to \(n\).
Specifically, $$ n! = n \cdot (n-1) \cdot (n-2) \cdot ⋯ \cdot 2 \cdot 1 $$ By convention, \(0! = 1 \).
The binomial coefficient \( \binom{n}{k} \) represents the number of ways to choose \(k\) elements from a set of \(n\) elements. In the context of the Binomial Theorem, it corresponds to the number of different ways to distribute the \(n\) powers of \(a\) and \(b\) in each term of the expansion.
Here's the Binomial Theorem applied to a few examples:
1. When \(n=2\): $$ (a + b)^2 = \binom{2}{0} a^2 b^0 + \binom{2}{1} a^1 b^1 + \binom{2}{2} a^0 b^2 = a^2 + 2ab + b^2 $$
2. When \(n=3\): $$ (a + b)^3 = \binom{3}{0} a^3 b^0 + \binom{3}{1} a^2 b^1 + \binom{3}{2} a^1 b^2 + \binom{3}{3} a^0 b^3 = a^3 + 3a^2b + 3ab^2 + b^3 $$
The Binomial Theorem has several important properties and applications, including:
- It provides an efficient method for expanding binomial expressions, particularly for large values of \(n\).
- It can be generalized to multinomials (expressions with more than two terms) and to negative or non-integer values of \(n\) using Newton's generalized binomial theorem.
- It can be used to derive various combinatorial identities, such as Pascal's identity and the hockey-stick identity.
- It has applications in probability theory, particularly in calculating probabilities for binomially distributed random variables.
The Binomial Theorem can also be understood in terms of its connection to the well-known Pascal's Triangle, which is an infinite triangular array of binomial coefficients. Each row of Pascal's Triangle corresponds to the coefficients of the binomial expansion of \((a+b)^n \) for increasing values of \(n\). The triangle starts with the first row, \(n=0\), and is constructed as follows: $$ \begin{array}{ccccccccccccccc} & & & & & & 1 & & & & & & \\ & & & & & 1 & & 1 & & & & & \\ & & & & 1 & & 2 & & 1 & & & & \\ & & & 1 & & 3 & & 3 & & 1 & & & \\ & & 1 & & 4 & & 6 & & 4 & & 1 & & \\ & 1 & & 5 & & 10 & & 10 & & 5 & & 1 \\ \end{array} $$ Each entry in Pascal's Triangle is obtained by summing the two numbers diagonally above it. For instance, the entry with value 6 in the fourth row is calculated by adding the two values above it (3 and 3).
Using Pascal's Triangle, you can quickly determine the coefficients of the binomial expansion without computing the binomial coefficients directly.
For example, the expansion of \( (a+b)^4 \) can be read from the fifth row of Pascal's Triangle: $$ a^4 + 4a^3 b+6a^2 b^2 + 4ab^3 + b^4 $$ The Binomial Theorem can also be extended to negative and non-integer exponents using the concept of infinite series. Newton's Generalized Binomial Theorem states that, for any real number \(r\) and any complex numbers \(a\) and \(b\) with \( |b| < |a| \) $$ (a+b)^r=\sum_{k=0}^{\infty} \binom{r}{k} b^k a^{r-k} $$ where the generalized binomial coefficient is defined as: $$ \binom{r}{k}=\frac{r!}{k!(r-k)!} $$ or $$ \binom{r}{k}=\frac{r(r-1)(r-2)⋯(r-k+1)}{k!} $$ The generalized binomial coefficients are used to compute the coefficients of the power series expansion of \( (a+b)^r \). This generalized theorem has numerous applications in calculus, such as finding the Taylor series of functions and solving differential equations.
In probability theory, the Binomial Theorem has applications in calculating probabilities for binomially distributed random variables. A binomial random variable represents the number of successes in a fixed number of Bernoulli trials, where each trial has only two possible outcomes (success or failure) and a constant probability of success.
The probability mass function of a binomial random variable \(X\) is given by: $$ P(X=k) = \binom{n}{k} p^k (1-p)^{n-k} $$ where \(n\) is the number of trials, \(k\) is the number of successes, and \(p\) is the probability of success on each trial. This formula directly employs the Binomial Theorem to calculate the probability of a specific outcome.