How does Pascal's triangle relate to binomial expansion?

1 Answer
Oct 25, 2015

Answer:

It tells you the coefficients of the terms.

Explanation:

Let's consider the #n-th# power of the binomial #(a+b)#, namely #(a+b)^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:

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

  • #(a+b)^3 = a^3 + 3a^2b + 3ab^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^2b# and #ab^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^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 #color(green)1#. And indeed, #(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.