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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Top NewestI 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 · 3 years ago
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-1<x<1 actually. – Bart Nikkelen · 2 years, 11 months ago
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Thank you, but do u know which ones only work for this case – Trevor Arashiro · 2 years, 11 months ago
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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, 9 months ago
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what is the derivation of 5th one. – Prashant Goyal · 3 years ago
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Standard result, it's related to "Newton's generalised Binomial theorem", but if you do want the derivation please see it here – Aditya Raut · 3 years ago
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Does the last one work for 0<x<1??? – Tasneem Khaled · 2 years, 11 months ago
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Definitely 2. – Eric Hernandez · 3 years ago
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set contains Everything – Gautam Sharma · 2 years, 11 months ago
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Is this for nerds like you? – Jack Daniel Zuñiga Cariño · 2 years, 11 months ago
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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. – Finn Hulse · 2 years, 11 months ago
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@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, 11 months ago
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Haha, anytime dude. :D – Finn Hulse · 2 years, 11 months ago
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Agreed. BTW, some nerds can be good at sports as well. – Sharky Kesa · 2 years, 11 months ago
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BTW sharky, participate in JOMO 8, we miss your submission.
@Sharky Kesa , JOMO 8 starts \(\color{Red}{\textbf{TOMORROW}}\) and has some good questions I made. – Aditya Raut · 2 years, 11 months ago
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(#Sharky_Surprises )
– Aditya Raut · 2 years, 11 months agoLog in to reply
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. @Jack Daniel Zuñiga Cariño – Aditya Raut · 2 years, 11 months ago
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