Let \(\mathrm{A}\) be the set of all "\(\color{Blue}{\text{useful}}\)" things.

Let \(\mathrm{B}\) be the set of all "\(\color{Green}{\text{Awesome}}\)" things.

Let \(\mathrm{C}\) be the set of all "\(\color{Red}{\text{fascinating}}\)" things.

Let \(\mathrm{D}\) be the set of all "\(\color{Brown}{\text{easily understandable}}\)" things.

What this note contains is an element of \(\mathrm{A \cap B \cap C \cap D}\)....

\[\color{Blue}{\textbf{Polynomial Expansions}}\]

\(\mathbf{1.}\quad \displaystyle \dfrac{1-x^{m+1}}{1-x} = 1+x+x^2+...+x^m = \sum_{k=0}^m x^k\)

\(\mathbf{2.} \quad \displaystyle \dfrac{1}{1-x} = 1+x+x^2+... = \sum_{k=0}^\infty x^k\)

\(\mathbf{3.}\quad \displaystyle (1+x)^n = 1+\binom{n}{1} x + \binom{n}{2}x^2+...+\binom{n}{n}x^n = \sum_{k=0}^n \dbinom{n}{k} x^k\)

\(\mathbf{4.}\quad \displaystyle (1-x^m)^n = 1-\binom{n}{1}x^m+\binom{n}{2}x^{2m}-...+(-1)^n\binom{n}{n} x^{nm}\)

\(\quad \quad \quad \quad\quad \displaystyle = \sum_{k=0}^n (-1)^n \dbinom{n}{k}x^{km}\)

\(\mathbf{5.}\quad \displaystyle\dfrac{1}{(1-x)^n} = 1+ \binom{1+n-1}{1} x + \binom{2+n-1}{2} x^2+...+\binom{r+n-1}{r} x^r+......\)

\(\quad \quad \quad \quad \quad \displaystyle =\sum_{k=0}^\infty \dbinom{k+n-1}{k} x^k \)

\(\color{Purple}{\text{Tremendously useful}}\) in calculating the \(\color{Blue}{\text{co-efficient}}\) of any term in specially generating functions that we come across, in many combinatorics problems...

Taken From - Alan Tucker's "Applied Combinatorics"

Good luck problem solving !

## Comments

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TopNewestI know that the second one only works for \(x<1\). But do any of the others work for only \(x>1\). Also, thank you so much for this note, it's very useful. – Trevor Arashiro · 2 years, 8 months ago

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– Bart Nikkelen · 2 years, 7 months ago

-1<x<1 actually.Log in to reply

– Trevor Arashiro · 2 years, 7 months ago

Thank you, but do u know which ones only work for this caseLog in to reply

Can you add parts of this page into the algebra wiki? I think that Algebraic Identities and Algebraic Manipulation - Identities, would be suitable places to add them. – Calvin Lin Staff · 2 years, 5 months ago

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what is the derivation of 5th one. – Prashant Goyal · 2 years, 7 months ago

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please see it here – Aditya Raut · 2 years, 7 months ago

Standard result, it's related to "Newton's generalised Binomial theorem", but if you do want the derivationLog in to reply

Does the last one work for 0<x<1??? – Tasneem Khaled · 2 years, 7 months ago

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Definitely 2. – Eric Hernandez · 2 years, 8 months ago

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set contains Everything – Gautam Sharma · 2 years, 7 months ago

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Is this for nerds like you? – Jack Daniel Zuñiga Cariño · 2 years, 7 months ago

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– Finn Hulse · 2 years, 7 months ago

Dude don't use nerds in a derogatory manner please. If you don't like nerds or aren't one yourself, you should remove yourself from Brilliant.org. Have a nice day.Log in to reply

@Finn Hulse , thanks for helping here, really !

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I truly like your comment, by these many likes :-

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@Cody Johnson :D – Aditya Raut · 2 years, 7 months ago

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– Finn Hulse · 2 years, 7 months ago

Haha, anytime dude. :DLog in to reply

– Sharky Kesa · 2 years, 7 months ago

Agreed. BTW, some nerds can be good at sports as well.Log in to reply

@Sharky Kesa , JOMO 8 starts \(\color{Red}{\textbf{TOMORROW}}\) and has some good questions I made. – Aditya Raut · 2 years, 7 months ago

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## (#Sharky_Surprises )

– Aditya Raut · 2 years, 7 months agoLog in to reply

@Jack Daniel Zuñiga Cariño – Aditya Raut · 2 years, 7 months ago

This is for using in generating functions we design for combinatorics problems.... For example, see the set "vegetable combinatorics".... (type in search bar simply).... That's for all who want to learn, nothing high-figh technique or anything, just formulas to get co-efficient of a specific term in a generating function.Log in to reply