Definition
Binomial Theorem (BT)
Definition
The Binomial Theorem is a fundamental theorem in algebra that describes the algebraic expansion of powers of a binomial. A binomial is an algebraic expression that contains two terms, which are usually joined by a plus or minus sign, such as $(a + b)$ or $(x - y)$. The Binomial Theorem provides a formula to expand expressions of the form $(a + b)^n$, where $n$ is a non-negative integer.
The theorem states that:
$$(a + b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k$$
Where $\binom{n}{k}$ is a binomial coefficient, also known as "n choose k", and it is given by:
$$\binom{n}{k} = \frac{n!}{k!(n-k)!}$$
Here, $n!$ denotes the factorial of $n$, which is the product of all positive integers up to $n$.
Binomial Coefficients
The binomial coefficients are the coefficients that appear in the binomial expansion. They have important combinatorial interpretations, representing the number of ways to choose $k$ elements from a set of $n$ elements without regard to the order.
Table of Differences and Important Points
| Property | Description |
|---|---|
| Symmetry | $\binom{n}{k} = \binom{n}{n-k}$ |
| Pascal's Rule | $\binom{n}{k} = \binom{n-1}{k-1} + \binom{n-1}{k}$ |
| Total Combinations | $\sum_{k=0}^{n} \binom{n}{k} = 2^n$ |
| Expansion Terms | The number of terms in the expansion of $(a + b)^n$ is $n+1$. |
Formulas
Binomial Expansion Formula: $$(a + b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k$$
Binomial Coefficient Formula: $$\binom{n}{k} = \frac{n!}{k!(n-k)!}$$
Factorial Definition: $$n! = n \times (n-1) \times (n-2) \times \cdots \times 2 \times 1$$
Examples
Expansion of $(a + b)^2$: $$(a + b)^2 = \binom{2}{0}a^2b^0 + \binom{2}{1}a^1b^1 + \binom{2}{2}a^0b^2 = a^2 + 2ab + b^2$$
Calculation of a Binomial Coefficient: $$\binom{5}{2} = \frac{5!}{2!(5-2)!} = \frac{5 \times 4 \times 3!}{2 \times 1 \times 3!} = \frac{20}{2} = 10$$
Application of Pascal's Rule: $$\binom{5}{3} = \binom{4}{2} + \binom{4}{3} = 6 + 4 = 10$$
Expansion of $(x - y)^3$: $$(x - y)^3 = \binom{3}{0}x^3(-y)^0 + \binom{3}{1}x^2(-y)^1 + \binom{3}{2}x^1(-y)^2 + \binom{3}{3}x^0(-y)^3 = x^3 - 3x^2y + 3xy^2 - y^3$$
The Binomial Theorem is not only useful in algebra but also in probability, statistics, and various fields of mathematics and science. Understanding the theorem and its applications is essential for solving a wide range of problems involving polynomial expansions.