%5En-binomial-expansion.jpeg "(a+b+c)^n Binomial Expansion")
The Binomial Theorem: (a+b+c)^n Expansion
The binomial theorem is a fundamental concept in algebra that provides a formula for expanding powers of a binomial expression. In this article, we will explore the expansion of (a+b+c)^n, where n is a positive integer.
Binomial Theorem Formula
The binomial theorem states that:
$(a+b)^n = \sum_{k=0}^n \binom{n}{k} a^{n-k} b^k$
where n is a positive integer, a and b are real numbers, and $\binom{n}{k}$ is the binomial coefficient.
Extension to Three Variables: (a+b+c)^n
Now, let's consider the expansion of (a+b+c)^n. Using the binomial theorem, we can extend the formula to three variables as follows:
$(a+b+c)^n = \sum_{i=0}^n \sum_{j=0}^{n-i} \binom{n}{i,j,n-i-j} a^{n-i-j} b^i c^j$
where i and j are non-negative integers such that i+j ≤ n.
Binomial Coefficients
The binomial coefficients $\binom{n}{i,j,n-i-j}$ can be calculated using the formula:
$\binom{n}{i,j,n-i-j} = \frac{n!}{i!j!(n-i-j)!}$
where n! is the factorial of n.
Example: (a+b+c)^3 Expansion
Let's consider the expansion of (a+b+c)^3. Using the formula above, we get:
$(a+b+c)^3 = a^3 + b^3 + c^3 + 3a^2b + 3a^2c + 3ab^2 + 3ac^2 + 3b^2c + 3bc^2 + 6abc$
Conclusion
In conclusion, the binomial theorem provides a powerful tool for expanding powers of binomial expressions. By extending the theorem to three variables, we can expand expressions of the form (a+b+c)^n. The binomial coefficients play a crucial role in this expansion, and can be calculated using the formula provided above.
