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The (a+b+c)^n Expansion: A Comprehensive Guide
Introduction
The (a+b+c)^n expansion is a fundamental concept in algebra and mathematics, used to expand the expression of a sum raised to a power. This concept is crucial in various mathematical fields, including combinatorics, probability, and statistics. In this article, we will delve into the world of (a+b+c)^n expansion, exploring its definition, formula, and examples.
Definition and Formula
The (a+b+c)^n expansion is defined as the expansion of the expression (a+b+c) raised to the power of n, where n is a positive integer. The formula for the expansion is given by:
$(a+b+c)^n = \sum_{k=0}^{n} \sum_{j=0}^{k} \binom{n}{k} \binom{k}{j} a^{n-k+j} b^{k-j} c^j$
where $\binom{n}{k}$ is the binomial coefficient, which represents the number of ways to choose k items from a set of n items.
Breaking Down the Formula
Let's break down the formula to understand it better:
- The outer summation runs from k=0 to n, where k represents the number of times the term b appears in the expansion.
- The inner summation runs from j=0 to k, where j represents the number of times the term c appears in the expansion.
- The term $\binom{n}{k}$ represents the number of ways to choose k items from a set of n items, which is used to calculate the coefficient of the term.
- The term $\binom{k}{j}$ represents the number of ways to choose j items from a set of k items, which is used to calculate the coefficient of the term.
- The terms $a^{n-k+j}$, $b^{k-j}$, and $c^j$ represent the powers of a, b, and c, respectively, in each term of the expansion.
Examples
Let's work through some examples to illustrate the (a+b+c)^n expansion:
Example 1: (a+b+c)^2
Using the formula, we get:
$(a+b+c)^2 = \sum_{k=0}^{2} \sum_{j=0}^{k} \binom{2}{k} \binom{k}{j} a^{2-k+j} b^{k-j} c^j$
Expanding the formula, we get:
$(a+b+c)^2 = a^2 + 2ab + b^2 + 2ac + 2bc + c^2$
Example 2: (a+b+c)^3
Using the formula, we get:
$(a+b+c)^3 = \sum_{k=0}^{3} \sum_{j=0}^{k} \binom{3}{k} \binom{k}{j} a^{3-k+j} b^{k-j} c^j$
Expanding the formula, we get:
$(a+b+c)^3 = a^3 + 3a^2b + 3a^2c + 3ab^2 + 6abc + 3ac^2 + b^3 + 3b^2c + 3bc^2 + c^3$
Conclusion
In conclusion, the (a+b+c)^n expansion is a powerful tool in mathematics, used to expand the expression of a sum raised to a power. By understanding the formula and working through examples, we can appreciate the beauty and complexity of this expansion. Whether you're a student or a professional, the (a+b+c)^n expansion is an essential concept to grasp in algebra and mathematics.
