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The Binomial Expansion of (a+b)^n
Introduction
The binomial expansion of (a+b)^n is a powerful tool in algebra and combinatorics. It allows us to expand an expression of the form (a+b)^n into a sum of terms, where each term is a product of powers of a and b. In this article, we will explore the binomial expansion of (a+b)^n and its applications.
The Binomial Theorem
The binomial theorem states that:
$(a+b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k$
where $\binom{n}{k}$ is the binomial coefficient, which can be calculated using the formula:
$\binom{n}{k} = \frac{n!}{k!(n-k)!}$
Example
Let's consider the expansion of (x+y)^3. Using the binomial theorem, we get:
$(x+y)^3 = \sum_{k=0}^{3} \binom{3}{k} x^{3-k} y^k$
Expanding the sum, we get:
$(x+y)^3 = \binom{3}{0} x^3 y^0 + \binom{3}{1} x^2 y^1 + \binom{3}{2} x^1 y^2 + \binom{3}{3} x^0 y^3$
Simplifying the expression, we get:
$(x+y)^3 = x^3 + 3x^2y + 3xy^2 + y^3$
Properties of Binomial Expansion
The binomial expansion of (a+b)^n has several useful properties:
Symmetry
The binomial expansion is symmetric, meaning that the coefficient of a^k b^(n-k) is the same as the coefficient of a^(n-k) b^k.
Pascal's Triangle
The binomial coefficients can be arranged in a triangle, known as Pascal's triangle. Each entry in the triangle is the sum of the two entries directly above it.
Applications
The binomial expansion has many applications in mathematics, physics, and computer science. It is used in counting and probability theory, as well as in algebra and calculus.
Conclusion
The binomial expansion of (a+b)^n is a powerful tool for expanding algebraic expressions. It has many applications in mathematics and other fields, and its properties, such as symmetry and Pascal's triangle, make it a fascinating topic to explore.
