Binomial theorem

A binomial is an algebraic expression that contains two terms. For example, $x+1$ is a binomial. The binomial theorem tells us how to expand an expression that has a binomial raised to any non-negative integer power. This is useful, because, for example, expanding $(x+1)^{20}$ manually is tedious. The theorem is useful in many areas of mathematics, including algebra and statistics.

Ideas behind the binomial theorem

Let's expand some powers of the generic binomial $a+b$, where $a$ and $b$ are real numbers.

$$(a+b)^0 = 1$$$$(a+b)^1 = a+b$$$$(a+b)^2 = (a+b) \cdot (a+b) = a^2 + 2ab + b^2$$$$(a+b)^3 = (a+b) \cdot (a+b)^2 =(a+b) \cdot (a^2 + 2ab + b^2) = a^3 +3a^2b+3ab^2+b^3$$We calculated the expansions using the distributive law of algebra.

Let's take a closer look at $(a+b)^3$:

$$(a+b)^3 = (a+b) \cdot (a+b) \cdot (a+b) = (a+b)(a+b)(a+b) = aaa + aab +aba+baa+ abb +bab + bba + bbb$$For an expansion in the form when "like terms" have not yet been collected, a term is found by taking the product of one component from each of the three factors. In other words, a term is $(a \text{ or } b) \cdot (a \text{ or } b) \cdot (a \text{ or } b)$. Two possible values can be selected from each of the three factors, so the total number of terms in the expansion is $2^3=8$. Here, we used arrangements with repetition to calculate the total number of terms in the expansion.

The "like terms" of the expansion can be combined. This means that the terms that have the same variable components can be added together. In the case of the expansion of $(a+b)^3$, for example, $aab$, $aba$, and $baa$ can be combined to give $3a^2 b$. There are ${3 \choose 1} = 3$ or ${3 \choose 2} = 3$ possible orderings of two $a$'s and one $b$, so the coefficient of $a^2b$ is $3 \choose1$. Likewise, there are $3 \choose 1$ or $3 \choose 2$ possible orderings of one $a$ and two $b$'s, resulting in the combined term $3ab^2$. Here, we used combinations to calculate the number of like terms.

With the above analysis in mind, the expansion after the "like terms" are combined can be written as follows.

$$(a+b)^3 = {3 \choose 0} a^3 + {3 \choose 1} a^2b + {3 \choose 2} ab^2 + {3 \choose 3} b^3$$

Binomial theorem

The binomial theorem generalizes the formula discussed in the previous section, stating:

$$(a+b)^n = \sum_{k=0}^{n} {n \choose k} a^{n-k}b^k$$

where $n$ is a non-negative integer. This is also known as the binomial identity.

The coefficients of the terms in the expanded expression are the combination ${n \choose k}$. Remember that

$${n \choose k} = {n! \over k!(n-k)!}$$As an example, let's expand $(a+b)^4$ using the binomial theorem.

$$(a+b)^4 = \sum_{k=0}^{4} {4 \choose k} a^{4-k}b^k$$

$$= {4\choose 0}a^4b^0 + {4\choose 1}a^3b^1 + {4\choose 2}a^2b^2 + {4\choose 3}a^1b^3 + {4\choose 4}a^0b^4$$$$= a^4 + 4a^3b + 6a^2b^2 + 4ab^3 + b^4$$

Likewise,

$$(a+b)^5 = \sum_{k=0}^{5} {5 \choose k} a^{5-k}b^k$$$$= {5\choose 0}a^5b^0 + {5\choose 1}a^4b^1 + {5\choose 2}a^3b^2 + {5\choose 3}a^2b^3 + {5\choose 4}a^1b^4 + {5\choose 5}a^0b^5$$$$= a^5 + 5a^4b + 10a^3b^2 + 10a^2b^3 + 5ab^4 + b^5$$

Conclusion

The expansion of a non-negative integer power $n$ of a binomial has $2^n$ total terms before the "like terms" are combined. The coefficient of the term after "like terms" are combined is calculated using combinations. These two observations give rise to the binomial theorem, which states that

$$(a+b)^n = \sum_{k=0}^{n} {n \choose k} a^{n-k}b^k$$