How does Pascal's triangle relate to binomial expansion?

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How does Pascal's triangle relate to binomial expansion?

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Answer 1 · 2015-10-25T10:30:21

It tells you the coefficients of the terms.

Explanation:

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

${(a + b)}^{2} = a^{2} + 2 a b + b^{2}$ $(a+b)^2 = a^2+2ab+b^2$ . All terms are of degree two: The exponent of $a^{2}$ $a^2$ is $2$ $2$ , and the same goes for $b^{2}$ $b^2$ , while in $2 a b$ $2ab$ , we have $a b = a^{1} b^{1}$ $ab=a^1b^1$ , and so again $1 + 1 = 2$ $1+1=2$ .
${(a + b)}^{3} = a^{3} + 3 a^{2} b + 3 a b^{2} + b^{3}$ $(a+b)^3 = a^3 + 3a^2b + 3ab^2+b^3$ , and all terms are either cubic ( $a^{3}$ $a^3$ and $b^{3}$ $b^3$ ), or the exponents of the variables sum up to three: $a^{2} b$ $a^2b$ and $a b^{2}$ $ab^2$ lead to $1 + 2 = 2 + 1 = 3$ $1+2=2+1=3$ .

So, when expanding the power of a binomial, you must count how many possible combinations you have to find numbers $i$ $i$ and $j$ $j$ such that $i + j = n$ $i+j=n$ . These numbers will be the exponents of the variables, and you will consider the sum of $a^{i} b^{j}$ $a^ib^j$ 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 $1$ $color(green)1$ . And indeed, ${(a + b)}^{0} = 1$ $(a+b)^0=color(green)1$ .

The second line is $color(green)1 \ \ color(red)1$ . And in fact, $(a+b)^1 = color(green)1a+ color(red)1 b$ .

The third line is $color(green)1 \ \ color(blue) 2\ \ color(red)1$ . And as we've seen above,
$(a+b)^2=color(green)1a^2 +\ \ color(blue) 2ab\ + color(red)1b^2$ .

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

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How does Pascal's triangle relate to binomial expansion?

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