%5En-formula.jpeg "(a+b+c)^n Formula")
The Formula for (a+b+c)^n
Introduction
The formula for (a+b+c)^n is a fundamental concept in algebra and is widely used in various mathematical disciplines. This formula is an extension of the binomial theorem, which deals with the expansion of powers of a binomial expression. In this article, we will explore the formula, its derivation, and some examples of its application.
The Formula
The formula for (a+b+c)^n is given by:
$(a+b+c)^n = \sum_{k=0}^n \sum_{i=0}^{k} \sum_{j=0}^{i} {n \choose k} {k \choose i} {i \choose j} a^{n-k} b^{k-i} c^{i-j}$
where ${n \choose k}$, ${k \choose i}$, and ${i \choose j}$ are the binomial coefficients.
Derivation
To derive this formula, we can start with the binomial theorem, which states that:
$(a+b)^n = \sum_{k=0}^n {n \choose k} a^{n-k} b^k$
Now, we can rewrite the expression (a+b+c)^n as:
$(a+b+c)^n = ((a+b)+c)^n$
Using the binomial theorem, we can expand the expression inside the parentheses:
$(a+b+c)^n = \sum_{k=0}^n {n \choose k} (a+b)^{n-k} c^k$
Next, we can expand the expression (a+b)^{n-k} using the binomial theorem again:
$(a+b)^{n-k} = \sum_{i=0}^{n-k} {n-k \choose i} a^{n-k-i} b^i$
Substituting this expression into the previous equation, we get:
$(a+b+c)^n = \sum_{k=0}^n {n \choose k} \sum_{i=0}^{n-k} {n-k \choose i} a^{n-k-i} b^i c^k$
Finally, we can rearrange the terms to get the final formula:
$(a+b+c)^n = \sum_{k=0}^n \sum_{i=0}^{k} \sum_{j=0}^{i} {n \choose k} {k \choose i} {i \choose j} a^{n-k} b^{k-i} c^{i-j}$
Examples
Example 1
Find the expansion of (a+b+c)^3.
Using the formula, we get:
$(a+b+c)^3 = a^3 + b^3 + c^3 + 3a^2b + 3a^2c + 3ab^2 + 3b^2c + 3ac^2 + 6abc$
Example 2
Find the expansion of (x+y+z)^4.
Using the formula, we get:
$(x+y+z)^4 = x^4 + y^4 + z^4 + 4x^3y + 4x^3z + 4xy^3 + 4y^3z + 4xz^3 + 6x^2y^2 + 6x^2z^2 + 6y^2z^2 + 12x^2yz + 12xy^2z + 12xz^2y + 24xyz^2$
Conclusion
In this article, we have derived and explored the formula for (a+b+c)^n. This formula is a powerful tool for expanding powers of a trinomial expression and has numerous applications in various fields of mathematics.
