Can we really do "-2 choose 5"? Generalized Binomial Theorem

Generalized binomial theorem. Can we really do "-2 choose 5"?
Here we will see how to extend the Binomial Theorem to any real number exponent, namely, we will be expanding (a+b)^r. I will also tell you why 1/(1-x) is the best friend!
Part 1: • How to expand (a+b)^n ...
part 2: • Can we really do "-2 c...
part 3: • the easy way to expand...
Bonus part: coming soon
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[@AndDiracisHisProphet]

yeah. sometimes when i have five options i feel like to choose minus two of those.

[@blackpenredpen]

Hahah, I do the same too!

[@Himineejimineee]

you mean that when you have -2 options, you choose 5 of them...

[@AndDiracisHisProphet]

@@Himineejimineee it's a joke

[@Himineejimineee]

​@@AndDiracisHisProphetI know! you just interpreted the notation backwards, that's all!

[@AndDiracisHisProphet]

@@Himineejimineee no. it was a joke

[@roderickwhitehead]

Every champion needs an entrance anthem! Epic.

[@jadegrace1312]

Using the limit test, the series converges if |b|

[@magnetonerd4553]

I remember when I first learned this technique (spent a lot of time reading Knuth back in the day). Has amazing applications to approximating inverse square laws for low orbits (small distances).

[@IISH4RKZzII]

This is so cool I just became a patreon and bought the best friend t-shirt😃😃

[@blackpenredpen]

Awwww thank you!!!

[@thenewguy7527]

Even though I didn't know anything here, I liked how you explained everything in detail!

[@illbeinyourdreams5259]

same cant wait to understund what he is saying

[@arlenestanton9955]

Yes, he is very enthusiastically through

[@barrowmeoct04]

So wait, I had 2 'cookies' in the Jar, My brother decides to eat 4 of them , leaving me -2 cookies I come home after work one day thinking " man, I could devour 5 cookies" How many ways of choosing 5 cookies do I have? -6

[@aimantajuararique3539]

yup got to love math

[@frozenturtl827]

yep

[@toqa002]

So grateful that is an another part! 💛

[@arequina]

I just realized I'm wearing the best friend shirt while watching this...whoaaah!

[@blackpenredpen]

Awww thank you!!!

[@dnpendown3199]

Well Your HairStyle is quite transformed over an Year sir😂😂😂, Thanks for the lessons!! Love from India🇮🇳

[@chaitanyagadekar5025]

When i was Joined first time this channel by ur integrals Problems...! At the first time i couldn't Understabd what ur talking. But now i can understand everything dude :)

[@pierreabbat6157]

There are -6 ways of choosing 5 objects from a set of -2. Are there also -6 ways of choosing -7 from a set of -2?

[@stephenbeale4765]

if it helps, you can just think of it as a taylor series in disguise, which is what it is

[@Yadobler]

@@stephenbeale4765 why is everything a taylor series in disguise! e^c? taylor series rCk? taylor series sin(x)? taylor series dflasnkfslkfjsdfkljlfjaflkdjlf

[@IvyANguyen]

Wow! This was so cool! Never knew negative numbers were possiblre in these. Are things like C(3,5) (where the 2nd number is higher than the 1st) really undefined or is there an actual answer to those, too?

[@blackpenredpen]

That will be just 0.

[@holyshit922]

Use it for expanding 1/sqrt(1-2xt+t^2) (Possibly need to use binomial theorem twice)

[@melissamiranda2525]

OMG That's awesome!! I love math every day more

[@habiboi_]

Came to this video completely clueless, now I feel like I have a pretty solid understanding of this! tysm

[@Lamiranta]

At the end of calculus I will have one best friend... That can save my life P.S. Thanks for a video about GBT. Waiting for Cantor set

[@honzaa6235]

dr peyam's appearance ( 3:59 ) was perfect ahah

[@Mystery_Biscuits]

Valid for abs(b/a) < 1

[@angelmendez-rivera351]

The condition is that |a| > |b|, but this can be reformulated so that one need never state this condition. Notice that for all pairs (a, b) in R^2, the equation a + b = max(a, b) + min(a, b) is true. Thus one can rewrite (a + b)^r = max(a, b)^r·[1 + min(a, b)/max(a, b)]^r, and now the series is bound to converge regardless of what the pair (a, b) is, since min(a, b)/max(a, b) is by definition less than 1 in absolute value.

[@karanpurohit26]

Awesome 😎

[@SuleymanKel]

why is a>b a condition?

[@angelmendez-rivera351]

@@SuleymanKel |a| > |b| is a condition because 1 > |b/a| is a condition. Otherwise, the series diverges, unless you rearrange the series to be in terms of |a/b| instead.

[@GaryTugan]

Awsome. And NOW I know what your Best Friend means :)

[@ronycb7168]

The falling factorial, generalised binomial coefficient .. seems interesting, thanks!

[@techgod360]

Could you please make a video on how to be better at trigonometry, please?

[@techgod360]

Thank you very much. You are the best KZbinr.

[@dadkinson]

The r choose k explanation came after the -2 choose 5 example, which felt inverted to me. Great video, though, thank you

[@ssdd9911]

(r)k can be r! if (negative integer)!/(negative integer)! is defined if the dividing by 0 parts can be cancelled out

[@kkenny]

Mr blackpenredpen, can you please make a video about the purpose or usefulness of calculus? I really need to know so that I can be more motivated to study Calculus 2. Thanks!

[@sensei9767]

Differential equations are really important in physics. Alot of things are derivatives: velocity, acceleration, density, ...

[@shahjahonsaidmurodov]

A lot of the early parts of calculus are mainly finding rates of change (and the rates of change in those rates in change), and in finding exact areas under curves. One real life use case of this that I can think of is with distance/time, speed/time, acceleration time graphs. The speed/time graph represents the change in distance The acceleration/time graph represents the change in speed The distance/time graph represents the area under a speed/time graph And the speed/time graph represents the area under an acceleration/time graph. Calculus will almost only be useful in this topic (kinematics) when those graphs are not straight lines (when they are curves). This is because with straight lines, the rate of change is just the gradient, and the area under the curve is just a triangle. Calculus is useful for the more realistic cases, like when acceleration is not constant (resulting in the speed/time graph being a curve, so you can’t perfectly work out the area underneath it using simple triangles).

[@zxf07]

Thanks.. From 🇧🇩

[@ThaSingularity]

If you like binomial coefficients, then you should check out Gaussian binomial coefficients.

[@davidwright8432]

Comment on several of the comments: people keep wondering what about complex numbers. My guess is that unless there's a 'super-generalization', 'n choose r' can't work for complex numbers because they aren't ordered the way integers are; there's no notion of a 'next bigger' or 'next smaller' complex number. I'd love to be proved wrong - because the solution would be mind-boggling - to my mind anyhow.

[@afernandesrp]

You emphasized so much on the exponent being a Real number that now I want to know about if it was a complex number. Can (a+b)^ni be done?

[@SuleymanKel]

yes

[@davidwright8432]

I need to watch the other videos! Just came on this cold. My initial guess was that you'd have (-2)!/(5)!(-7)!. , simply using the 'restricted' binomial coefficient form. Then the second term in the generalized denominator vanished, but in the end, reappeared! This is the first time my brain has hurt after any of your excellent videos. Memo to self: start at beginning, not middle! (I already knew the 'traditional' 'n choose r' formula.) the difference between us: you knew what you were doing; I didn't. But I'll work on it1 thanks.

[@jafecc]

Here, -2 choose k = (-1)^k * (k+1) [k is an integer].

[@shahsamir4442]

Please please tell me why infinite terms? I didn't understand?

[@gamingguys6571]

Try this problem, Cot^4(x) - Cot^2(x)= 1 Cos^4(x)+ Cos^2(x) = ?

[@stephenbeale4765]

1 + cot^2x = cot^4x cosec^2x = cot^4x 1/sin^2 x= cos^4x/sin^4x 1 = cos^4x/sin^2x cos^4 x = sin^2 x cos^4 x = 1-cos^2x cos^4x + cos^2x = 1

[@bachirblackers7299]

7:35 falling factorials or falling powers . Another annotation like powers anderlined .

[@Nickesponja]

Now do it for trinomials

[@roderickwhitehead]

See video description for link to Part 3.

[@lilyyy411]

Rip... It doesn't work for complex numbers or does it?

[@blackpenredpen]

Icestrike411 Official it does!!!

[@lilyyy411]

@@blackpenredpen :thinking: we need a video on that. (I know you did it with a different method before)

[@danielgates7559]

Icestrike411 Official search up Laurent series... I do agree with you that complex analysis videos are going to be fun.

[@marcushendriksen8415]

Fun fact: merely summing the binomial coefficients for r=1/2 gets you the square root of 2!

[@SuleymanKel]

in which case?

[@marcushendriksen8415]

@@SuleymanKel in which case what? It's a fun fact dude, there's nothing deep going on here 🤣

[@SuleymanKel]

r=1/2 for which equation exactly? @@marcushendriksen8415

[@user-bf7zo1lh1z]

if the power is zero don't you need to add the condition that a is not equal to negative b?

[@pbj4184]

It is assumed the viewer would know that

[@beri4138]

How do I solve: Find the coefficient of x^12 in the expression: (x^5+x^6+x^7+...)^2 * (1+x+x^2+...)^8

[@GreenMeansGOF]

aCb=Γ(a+1)/(Γ(b+1)*Γ(a-b+1)) for all real (or maybe complex😳) numbers?

[@abdelalibouchouicha1683]

Thanks

[@blblbl2750]

it will converge if |b| < |a| and diverge if |b| > |a| (ratio test), not sure about the case |a| = |b|..

[@呂永志-x7o]

教學建議：首先應該計算正整數時結果不會改變。製作T恤的建議：將公式寫在正面，弄成鏡像，我會喜歡這樣自拍。

[@blackpenredpen]

hahahaha! 我其實正在弄Laplace Transform的T呢, 然後是正面的!!! (老師專用版)

[@filipjansson8885]

Why do we not put K! x (r-K)! On the denominator?

[@roderickwhitehead]

I wondered the same thing... and I was too lazy to think about it.

[@MrGiancarlo36]

14:32 isn't -2 ?

[@TrimutiusToo]

But with best friend you can raise 1 to the k and (-x) to the r-k and get a different expansion...

[@user-wu8yq1rb9t]

Thank you so much Dear Teacher *bP🖋️rP🖍️* ❤️

[@MmmM-mf3zd]

I still remember how my mind just blew up learning this theoremal and then use it to get the x^n derivated by definition... I feel so powerful xdd

[@nalat1suket4nk0]

Why don't you check if it is right by computing 1/(a+b)^2

[@1suoregnad]

thank you

[@yajurphullera9396]

Do generating functions next.

[@mauricepanero]

Surely if a>b you will get convergence?

[@jadegrace1312]

|b|

[@SuleymanKel]

why is the condition a>b? I don't understand

[@DiegoMathemagician]

Now generalize (a+b+c+...)^n (multinomial theorem)

[@janernn]

So here's the deal

[@epic0tom113]

Any reason why this one isn’t public?

[@blackpenredpen]

To make the people who actually ready my description special. : ) Thank you!

[@epic0tom113]

blackpenredpen you’re the one who deserves the thanks 😁

[@timeonly1401]

It's a closely-guarded secret. Now we'll have to kill everyone who watched this video. ;-)

[@dipanjandutta4963]

How to expand fractional power??...Sir can you please make a video on such a topic...

[@Engr.vin9520]

what if you are going to find the n term of binomial

[@timka3244]

Solve the Product intregal (intregal to p f(x) dx) (If you can)

[@lvc6717]

Hey! Nice video! I was wondering what text font did you use in your pdf related rates doc from a few weeks ago?

[@dhunt6618]

Which is worse: bark(1/(1-x)) or bite(1/(1-x))?

[@muhammadqasim7056]

Factorials of fractions?cycloids?.Love the videos

[@chidielijahnwakpa9482]

Pretty cool, but a bit confused where you expressed r! as r(r-1)(r-2)...[r-(k-1)]. Isn't it not meant to be r! = r(r-1)(r-2)...[r-(k-1)][r-(k-1)-1][r-(k-1)-2]...3•2•1, which can again be written as r! = r(r-1)(r-2)...[r-(k-1)](r-k)! ? Thanks, best regards from Nigeria.

[@ssdd9911]

so this is what he meant when my teacher says this works for negatives and fractions

[@mooncake8320]

I was wondering, what happens if you evaluate a number for a root?

[@somanathdash7979]

Y are a legend in maths Like if u agree

[@kutuboxbayzan5967]

Prove with thailer series

[@SlipperyTeeth]

Is the Generalized Multinomial Theorem next?

[@talhaashraf6332]

can someone tell me why its the sum to infinity

[@manuelocana8074]

doesn't a need to be >0?

[@_P_a_o_l_o_]

Just out of curiosity, how do you prove that the formula is valid? To me, it looked like you pulled it out of the hat, so I would be happy to see a derivation of the formula

[@willnewman9783]

You can take the taylor series for (1+x)^r, and note that it is what the video days for a=1, b=x. Then, write x=a/b, and multiply through by b^r to get the formula for (a+b)^r.

[@timeonly1401]

Nevermind the proof; I want to lay my hands on that hat!!

[@_P_a_o_l_o_]

will newman Thank you very much, I hadn’t thought of that! Very neat

[@kiranbarnwal8503]

@blackpenredpen Hey bro! Please tell condition on a and b

[@newtonnewtonnewton1587]

Nice video

[@marcushendriksen8415]

If b and r are both negative, does that mean its binomial expansion will have all its terms positive?

[@marcushendriksen8415]

The reason I think so is that every b term with an odd degree will be negative, and its corresponding coefficient will have an odd number of negative factors, thus the whole thing should be positive.

[@andreapaps]

Would the condition be abs(b/a)

[@davidloomis6668]

Could you solve int5/1 e+4x^5+3x^2 dx

[@jelojavier5393]

May I ask a 'dumb' question, is the constant term in polynomials a coefficient? This question is for the sum of the coefficients for polynomial expansion with a constant term. Thanks!

[@devianawijaya7735]

why for n /in/mathbb{Z^{+}}, a interval for values of x so that the expansion will be valid must be applied?

[@loukafortin6225]

Is there a binomial theorem for roots?

[@whyit487]

*Why was the notification so late!*

[@holychurchtuni4073]

super

[@shuddhoshawttoroy6257]

Is the proffesor/teacher, explaining this topic from South Korea?

[@lumina_]

no...

[@Pineapplecake]

Sorry, I already wore the best friend shirt to my series test.

[@dmytro_shum]

wow, you are reading my mind (the minus power of binom)

[@slasher736]

3:45 😂😂

[@guidichris]

Wouldn't you need to get the person in front of you to wear the tshirt on test day? Or I guess you could wear the shirt backwards.

[@abdelhamid_moukrime]

it is true for the number complex to check it out en.wikipedia.org/wiki/Binomial_theorem

[@papapapapapapageno]

1-x > 0 iff |x| 0?

[@potrinhoconfuso]

1 - |x| > 0 *

[@siwi5315]

Could you do x^x=a (a element of real (or complex numbers)) and show a solution

[@angelmendez-rivera351]

siwi He has already

[@siwi5315]

@@angelmendez-rivera351 i cant find it then...could you help me with the link or Name of the Video?...and thanks btw

[@timgoppelsroeder121]

Why cant r be complex?

[@fluffymassacre2918]

Damn

[@msolec2000]

That's swell, but also, people should know this: a^(-2) + b^(-2) = h^(-2)

[@kristoferkenser5673]

Sorry, but can anyone explain me from where he takes K? Is it totally random?

[@JianJiaHe]

Now generalize it to complex numbers

[@abdelhamid_moukrime]

it is true for the number complex to check it out en.wikipedia.org/wiki/Binomial_theorem

[@AlessandroZir]

👏👏👏👏

[@_kareem202]

Can you intgraetion the root of cosx

[@gabriel78493]

Y ~ U(x,1)

[@ThePharphis]

what are these sets? Or something else?

[@sarthakbindal6058]

Sir! I really love the way to make maths easy....you are so interesting and makes maths more interesting!! I want to request you something! That can you please start your next videos with "NAMASTE" . Love💓 from india🇮🇳 Please please do so😄😄😙

[@antarikshkumar6194]

did someone notice that the integral 'best friend.dx' is ln(best friend)+c.

[@jordanrutledge7943]

isn't it negative ln(best friend)+c

[@antarikshkumar6194]

@@jordanrutledge7943 it will be negative ln(1-x). if you take out a minus and reciprocal the log. you will get ln(1/1-x). i.e. ln best friend