Pascal,Triangle,and,the,Binomi education Pascal’s Triangle and the Binomial Expansion

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Expanding brackets can be a nightmare; even something like (a+b)^4 can takeages to multiply out and simplify. In this article, I explain how Pascal'striangle 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 bracketsthe slow way - if you do not know how to do this, then the article contains alink to a page explaining this.We will look at the expansions of (a+b)^2, (a+b)^3 and (a+b)^4 to try tospot 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^2Next, we must try to expand (a+b)^3. This is a little more complicated, butdon'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), andchange them into a^2 + 2ab + b^2. This leaves one more bracket of (a+b) tomultiply 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, andadd all the stuff we get together, we will have expanded (a+b)^3!a^2 + 2ab + b^2Multiplied by a: a^3 + 2a^2b + b^2aMultiplied by b: a^2b + 2ab^2 + b^3So (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^3I don't know if you'd agree, but this is taking forever - I just want to geton with the interesting stuff and show you how all this links in with Pascal'striangle! Therefore, rather than spend ages working through (a+b)^4 and (a+b)^5with you, I will just tell you what they are. I expect you will be only toohappy to take my word for it about these expressions, but if you don't believeme, 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^5One pattern you can see here is that the powers of a decrease from 5 and thepowers of b increase to 5 as you work your way from left to right along theexpression. However, what is the pattern in the coefficients (a posh name forthe numbers in front of the powers of a and b, shown in bold)? They are thenumbers from Pascal's triangle! (Notice how I have put 1s in from of the firstand 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 dothis.)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 knowledgeof the patterns in the binomial expansion and with Pascal's triangle in frontof 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 willdecrease from 10, and the powers of b will increase to 10 as we work our wayalong the expression. So, not worrying about the coefficients for a second, thepowers 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^10Next, we need to use the 10th row of Pascal's triangle to get thecoefficients:1, 10, 45, 120, 210, 252, 210, 120, 45, 10, 1Finally, we just bung these numbers in front of the powers of a and b shownabove 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^10It is clear, therefore, that Pascal's triangle can save you ages of hardwork when multiplying out brackets. You probably never even dreamt that youcould expand such a hideous expression as (a+b)^10. Furthermore, a calculatorcouldn't expand brackets like this (well, you're never quite sure whatscientific calculators can do these days), so it is definitely a good idea toadd a copy of Pascal's triangle to the other mathematical weaponry in yourpencil case! 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