How does Pascal's triangle relate to binomial expansion?

Oct 25, 2015

It tells you the coefficients of the terms.

Explanation:

Let's consider the $n - t h$ power of the binomial $\left(a + b\right)$, namely ${\left(a + b\right)}^{n}$. It must be a polynomial in $a$ and $b$ of degree $n$, and so every term must be of degree $n$, which means that the exponents of $a$ and $b$ must sum to $n$. Let's make a couple of examples:

• ${\left(a + b\right)}^{2} = {a}^{2} + 2 a b + {b}^{2}$. All terms are of degree two: The exponent of ${a}^{2}$ is $2$, and the same goes for ${b}^{2}$, while in $2 a b$, we have $a b = {a}^{1} {b}^{1}$, and so again $1 + 1 = 2$.

• ${\left(a + b\right)}^{3} = {a}^{3} + 3 {a}^{2} b + 3 a {b}^{2} + {b}^{3}$, and all terms are either cubic (${a}^{3}$ and ${b}^{3}$), or the exponents of the variables sum up to three: ${a}^{2} b$ and $a {b}^{2}$ lead to $1 + 2 = 2 + 1 = 3$.

So, when expanding the power of a binomial, you must count how many possible combinations you have to find numbers $i$ and $j$ such that $i + j = n$. These numbers will be the exponents of the variables, and you will consider the sum of ${a}^{i} {b}^{j}$ with some coefficients. And here comes Pascal's triangle. It tells you the coefficients of the progressive terms in the expansions.

For example, the first line of the triangle is a simple $\textcolor{g r e e n}{1}$. And indeed, ${\left(a + b\right)}^{0} = \textcolor{g r e e n}{1}$.

The second line is $\textcolor{g r e e n}{1} \setminus \setminus \textcolor{red}{1}$. And in fact, ${\left(a + b\right)}^{1} = \textcolor{g r e e n}{1} a + \textcolor{red}{1} b$.

The third line is $\textcolor{g r e e n}{1} \setminus \setminus \textcolor{b l u e}{2} \setminus \setminus \textcolor{red}{1}$. And as we've seen above,
${\left(a + b\right)}^{2} = \textcolor{g r e e n}{1} {a}^{2} + \setminus \setminus \textcolor{b l u e}{2} a b \setminus \setminus + \textcolor{red}{1} {b}^{2}$.

And so on: if you look above, you have that the coefficients of the cubic expansion are $1 \setminus \setminus 3 \setminus \setminus 3 \setminus \setminus 1$, which is exactly the fourth line of the triangle.