Proof,the,Link,Between,Pascal, education Proof of the Link Between Pascal’s Triangle and the Binomial

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A famous use of Pascal's triangle is in the Binomial Expansion, themultiplication and simplification of brackets. Less well known, however, is thereason why this intriguing link between these two seemingly unrelated areas ofmathematics actually exists. As the binomial expansion deals with multiplyingout brackets algebraically, you might think that a proof of its link withPascal's triangle would also involve a lot a complicated algebra. In thisarticle, however, I have attempted to give an informal proof of this link usingas little algebra as possible.As with many proofs related with Pascal's triangle, the link with theBinomial expansion can be proved inductively. However, in this article, we'renot going to worry too much about what proof by induction is, or use any of thenotation or terminology that often comes with it. I am just going to explainthe rationale behind the expansion of (a+b)^6 being the 6th row of Pascal'striangle, and from there, it should be clear how my arguments can begeneralised to cover any power of (a+b).Right then, let's think about (a+b)^6. It can be made from the expansion of(a+b)^5 all multiplied by another bracket of (a+b), as shown below:(a+b)^6 = (a+b)(a+b)(a+b)(a+b)(a+b)(a+b) = (a+b)^5(a+b)= (a^5 + 5a^4b + 10a^3b^2 + 10a^2b^3 + 5ab^4 + b^5)(a+b)= a(a^5 + 5a^4b + 10a^3b^2 + 10a^2b^3 + 5ab^4 + b^5) + b(a^5+ 5a^4b + 10a^3b^2 + 10a^2b^3 + 5ab^4 + b^5)This is looking complicated and not very revealing. However, we can nowstart thinking about how all this links in with Pascal's triangle. As anexample, say we are trying to make a^2b^4. As shown in the expression above, toget the terms of (a+b)^6, we can multiply a^5 + 5a^4b + 10a^3b^2 + 10a^2b^3 +5ab^4 + b^5 by either a or b. This leaves uswith two ways of achieving our goal of getting a^2b^4. We could choose a^2b^3and multiply it by b from outside the bracket, or we couldstart with ab^4 and multiply it the a from outside thebracket.As these are the only two ways, the coefficient of a^2b^4 is the sum of thecoefficients of ab^4 and a^2b^3. Basically, the number of lots of a^2b^4 I endup with is the number of lots of a^2b^3 from the bracket there are, plus thenumber of lots of ab^4 there are. If you look at a^5 + 5a^4b + 10a^3b^2 +10a^2b^3 + 5ab^4 + b^5, you will see that we have 10a^3b^2 and 5ab^4, so wewill end up with 15 lots of a^2b^4.You may now be able to see where this is leading. We just need to thinkabout where these powers are supposed to be represented in Pascal's triangle.10a^2b^3 and 5ab^4 are both from the expansion of (a+b)^5, so they will be inthe 5th row of Pascal's triangle. In the claim we are trying to prove we alsostate that you add one to the power of b to see how many numbers in you have tocount to get your desired coefficient. So the coefficient of a^2b^3 will berepresented by the 4th number along, and ab^4 by the 5th number along. Finally,we can say that 15a^2b^4 will be the 5th number in on the 6th row. If you thinkabout it, these three numbers are positioned such that they will form a littletriangle of numbers in Pascal's triangle.Applying these arguments to any situation, we can see that the rules ofPascal's triangle will hold for any term from any power of (a+b), as we canalways split our term into the coefficients of two separate terms positionedadjacently directly and directly above our initial term. The only time when our"term-splitting" tactic won't work is for (a+b)^1, but (a+b)^1 = 1a +1b, and 1,1 is the first row of Pascal's triangle, so this obviously worksanyway.We're done! This was not an easy proof, so congratulations for getting thisfar! Even if you did not fully understand all of it, hopefully it has given yousome insight into how the seemingly unrelated topic of expanding and thebinomial expansion is in fact so closely linked to Pascal's triangle. Normal 0 false false false EN-GB X-NONE X-NONE /* Style Definitions */ table.MsoNormalTable{mso-style-name:"Table Normal";mso-tstyle-rowband-size:0;mso-tstyle-colband-size:0;mso-style-noshow:yes;mso-style-priority:99;mso-style-qformat:yes;mso-style-parent:"";mso-padding-alt:0cm 5.4pt 0cm 5.4pt;mso-para-margin-top:0cm;mso-para-margin-right:0cm;mso-para-margin-bottom:10.0pt;mso-para-margin-left:0cm;line-height:115%;mso-pagination:widow-orphan;font-size:11.0pt;font-family:"Calibri","sans-serif";mso-ascii-font-family:Calibri;mso-ascii-theme-font:minor-latin;mso-fareast-font-family:"Times New Roman";mso-fareast-theme-font:minor-fareast;mso-hansi-font-family:Calibri;mso-hansi-theme-font:minor-latin;mso-bidi-font-family:"Times New Roman";mso-bidi-theme-font:minor-bidi;}