[Discrete Mathematics] Coefficient Extraction Examples

I do some questions on coefficient extraction from binomials. They are more complicated than your typical examples.

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[Shannxy]

Awman 2 days til my final and I've spent a total of 8 days trying to learn a whole semester's worth of discrete 1&2 maths.

I've just been rummaging through old exams though, and your videos have helped a lot!

[jessstuart]

Numerical answers...

[x^15] = 9080
[x^83] = 118360

I like to make a table when solving these. It helps me not to inadvertently drop a -1.

[Sui_Generis]

Great video, thanks for the explanations, in terms of complexity what could be studied passed on this topic as I am looking for something more challenging

[ArsenijeRadenovic]

Hello again Trev, i have a question regarding generating functions and coeff extraction:
In how many ways i can pick 21 cents if i have: 6 coins worth 1 cent, 5 coins worth 2 cents, 4 coins worth 5 cents?
What do i do when i get their generating functions like these:
(1-x^7)/(1-x) * (1-x^12)/(1-x^2) * (1-x^20)/(1-x^5) ? I need to find coef of x^21, should i multiply all of these equations and then use binomal formula or is there another way to solve this? Thanks a lot for all of your videos and your time spent to make them.

[Isaac]

I've been searching online about the way you alternate the sign of the coefficient whenever you have 1/(1-x)^n and I can't find why. You might need to check your formulae because I have a couple of books that describe:
1/(1-x)^n as the sum from i=0 to infinity of (n+i-1 choose i)x^i

and
1/(1+x)^n as the sum from i=0 to infinity of (-1)^i(n+i-1 choose i)x^i

So, I think you got those 2 the other way around. Please check this as your videos have been really helpful but this is throwing me off completely as I've been studying using both your videos and Grimaldi's "Discrete and combinatorial mathematics" where the previously mentioned identities are listed.
These identities make sense as they come from the simpler ones where the
sign for 1/1-x does not alternate and for 1/1+x it does alternate, then
you apply the Binomial Theorem.

[anvi.gupta]

Are there any better ways to do other than taking cases ?

[poopking]

Hey Trev,

at 9:16 you remove the -1 since it cancels out somehow. Could you explain with what it cancels out with to me?

[quinndougherty]

every time you do (-1)^11 (or some odd number) I don't know how it cancels out.

[pacman]

Hey Trev how would you do something like find the coef. of x^21of 1/(1-x) * 1/(1-x²)? Thanks for your videos.

[davemartin]

I hope MATLAB and NUMPY know how to do these operations.

[yasmin]

I don't understand how you choose 15 out of -4 in the first example

[xxx]

In the second example, why is the denominator (1-x^3)^10 and not (1-x)^10.
Because the original formula is (1-x^n+1)/(1-x)?

Also please reply ASAP. I have an exam tomorrow so...