Pascal’s Triangle and the Binomial Expansion

Expanding brackets can be a nightmare; even something like (a+b)^4 can take ages to multiply out and simplify. In this article, I explain how Pascal's triangle can come to the rescue by helping to expand powers of (a+b) as high as (a+b)^10 in a matter of seconds, not hours.

Please note that this article assumes knowledge of how to expand brackets the slow way - if you do not know how to do this, then the article contains a link to a page explaining this.

We will look at the expansions of (a+b)^2, (a+b)^3 and (a+b)^4 to try to spot the pattern of how expanding brackets is linked with Pascal's triangle.

(a+b)^2 = a^2 + ab + ba + b^2 = a^2 + 2ab + b^2

Next, we must try to expand (a+b)^3. This is a little more complicated, but don't panic! We can use what we have already found out to help us:

(a+b)^3 = (a+b)(a+b)(a+b)

= (a^2 + 2ab + b^2)(a+b)

All we have done above is take two of the three brackets of (a+b), and change them into a^2 + 2ab + b^2. This leaves one more bracket of (a+b) to multiply the whole of a^2 +2ab + b^2 by, hence giving us (a^2 + 2ab + b^2)(a+b). Now, if we mulitply everything in a^2 +2ab + b^2 by a and by b, and add all the stuff we get together, we will have expanded (a+b)^3!

a^2 + 2ab + b^2

Multiplied by a: a^3 + 2a^2b + b^2a

Multiplied by b: a^2b + 2ab^2 + b^3

So (a^2 + 2ab + b^2)(a+b) = a^3 + 2a^2b + b^2a + a^2b + 2ab^2 + b^3

= a^3 + 3a^2b + 3ab^2 + b^3

I don't know if you'd agree, but this is taking forever - I just want to get on with the interesting stuff and show you how all this links in with Pascal's triangle! Therefore, rather than spend ages working through (a+b)^4 and (a+b)^5 with you, I will just tell you what they are. I expect you will be only too happy to take my word for it about these expressions, but if you don't believe me, you can always check them yourself! All you would have to do to expand (a+b)^4 is take our answer for (a+b)^3 and multiply it by another bracket of (a+b), and then for (a+b)^5 multiply your answer by a further bracket of (a+b). Here they are then, along with (a+b)^1, (a+b)^2 and (a+b)^3:

(a+b)^1 = 1a + 1b

(a+b)^2 = 1a^2 + 2ab + 1b^2

(a+b)^3 = 1a^3 + 3a^2b + 3ab^2 + 1b^3

(a+b)^4 = 1a^4 + 4a^3b + 6a^2b^2 + 4ab^3 + 1b^4

(a+b)^5 = 1a^5 + 5a^4b + 10a^3b^2 + 10a^2b^3 + 5ab^4 + 1b^5

One pattern you can see here is that the powers of a decrease from 5 and the powers of b increase to 5 as you work your way from left to right along the expression. However, what is the pattern in the coefficients (a posh name for the numbers in front of the powers of a and b, shown in bold)? They are the numbers from Pascal's triangle! (Notice how I have put 1s in from of the first and last terms of each expression, to make the pattern easier to spot. Multiplying by 1 does not change the value of anything, so I'm allowed to do this.)

This is very useful. Before, if I asked you to multiply out (a+b)^10 then you would have thought I was mad. Now, however, armed with knowledge of the patterns in the binomial expansion and with Pascal's triangle in front of you, you could expand (a+b)^10 in a matter of seconds, not hours. We will do (a+b)^10 together so I can show you how quick it is:

Firstly, as we are raising (a+b) to the power of 10, the powers of a will decrease from 10, and the powers of b will increase to 10 as we work our way along the expression. So, not worrying about the coefficients for a second, the powers of a and b we are dealing with are shown below:

a^10, a^9 b, a^8 b^2, a^7 b^3, a^6 b^4, a^5 b^5, a^4 b^6, a^3 b^7, a^2 b^8, ab^9, b^10

Next, we need to use the 10th row of Pascal's triangle to get the coefficients:

1, 10, 45, 120, 210, 252, 210, 120, 45, 10, 1

Finally, we just bung these numbers in front of the powers of a and b shown above to get our completed expansion:

(a+b)^10 = a^10 + 10a^9 b + 45a^8 b^2 + 120a^7 b^3 + 210a^6 b^4 + 252a^5 b^5 + 210a^4 b^6 + 120a^3 b^7 + 45a^2 b^8 + 10ab^9 + b^10

It is clear, therefore, that Pascal's triangle can save you ages of hard work when multiplying out brackets. You probably never even dreamt that you could expand such a hideous expression as (a+b)^10. Furthermore, a calculator couldn't expand brackets like this (well, you're never quite sure what scientific calculators can do these days), so it is definitely a good idea to add a copy of Pascal's triangle to the other mathematical weaponry in your pencil case!

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