#binomial coefficient

[blake447]

So here's an interesting fact I've learned while researching these things called Wolstenholme Primes. See they're related to the binomial coefficient and thus pascals triangle. We've probably all seen sierpinski's triangle in here (if not circle all the odd numbers). But here's something else interesting. If you look at the triangle, you'll see that for prime numbers, every element is divisible by that prime number. Look at 11 for example.

55, 165, 330, and 462 are all divisible by 11. This has to do with the binomial coefficient definition, in which you'll see that if unless we choose either all the elements, or none of them, the factor of our prime number is guaranteed not to cancel out from the numerator's factorial.

We can then use the fact that the sum of all the elements in the nth row of pascal's triangle adds up to 2^n. Once we remove the 1's on each end of the triangle (hence the minus 2) we know that for prime numbers p that all elements are divisible by p, their sum is as well!

Now, I don't know off the top my head if this is only true for prime numbers. It isn't immediately obvious imo. However, if this is strong enough to be an if and only if, then I think I have independently discovered perhaps the *worst* way to test for primality lmao, scaling exponentially instead of the standard logarithmically.

[larry-is-my-anchor1]

In which brainy freshman Stiles Stilinski wants star quarterback Derek Hale to join the math team, AKA math nerds in love.

[suchasourwolfie]

Sterek HS AU’s

Practice Makes Perfect Series by @blacktofade

One of the best Sterek HS AUs.  Derek is Captain of the Lacrosse team.  Scott and Stiles make the team somehow.  

Option C) Some Bad Guys are Werewolves, but not all Werewolves are Bad Guys by Calrissian18 @wellhalesbells​

All the high school days nostalgia Sterek style. QuietBroodingJock!Derek / Nerd!Stiles

Binomial Coefficients by @devildoll

So many beautifully awkward moments. Captured flawlessly the nerd / jock romance.

I laughed out loud at so many lines, but this is my favorite line: “Derek always looks slightly annoyed with everyone around him, and he still has more friends than Stiles.”

Here’s To You, Kiddo by MellytheHun @loserchildhotpants​

Jock!Werewolf!Protective!Pining!Derek, Artist!Stiles, Bully!Jackson. Sweet and easy HS AU read. 

What Good Are Rules If You Can’t Break Them by wishingonalightningbolt

Stiles and Derek use their antagonistic relationship to keep their no-strings-attached-sex relationship a secret. “It goes about as well as you’d expect.” 

The Socioeconomic Repercussions of Mutually Assured Destruction by alocalband

Derek is the cool, but quiet kid.  Stiles is a flaily nerd. Checks all the Sterek HS AU boxes. 

Tutor!Verse by betp @lavenderek​

Snack size vignettes following Jock!Derek and “Geek”!Stiles. The first Not Another Sterek Romance ... is completely fulfilling on it’s own if you are not looking for a series read. 

The Nerd Party by bibliosexual @biliosexxual

Lacrosse Captain (and closet nerd) Derek Hale and Unabashed nerd Stiles Stilinski discover they have a common interest and something to talk about. Hales!Live. 

The Athlete & the Criminal by @damnfancyscotch 

Well done spin on The Breakfast Club with Stiles as The Jock" and Derek as "The Criminal". Nothing too deep, but a quick feel good read.

Covelant Bonds Series by @halffizzbin 

Nerd!Derek and Jock!Stiles get paired as chemistry partners. 

You Were Only 17 Series by MellytheHun @loserchildhotpants​

Antagonistic!Derek “bullies” Stiles until the tension gets so thick a kiss is stolen and everything becomes clear. (The “bullying” in this fic is akin to some vexing pigtail pulling - attention seeking, not malicious.)  

You’ve Got Notes by the_gramophone

Jock!Derek, Nerd!Stiles. Derek and Laura are twins and in the same grade as Stiles. Stiles keeps receiving notes in his locker that are anonymous love letters. It’s sweet, then heart breaking, then brilliantly, dramatically butterfly inducing. 

You Can Hear It In the Silence by Emela

I’m sorry to see this one was deleted. It’s a bit more of a serious take on the HS AU with a “John Goode YA feel” about it. A secret relationship between a formerly popular Derek - who has been “outed” by a vindictive Kate and ostracized by a homophobic student body and a firmly closeted, newly popular Stiles.  Beyond the Sterek relationship it took on some coming of age themes in a societal context. 

[christinesficrecs]

Any smitten Stiles fic recs? I just love when Stiles is completely gone on Derek and I’m craving some good ones.

Sure! Here are a few. ❤️

Operation: Chick Flick by Inell | 7.3K

Stiles knows agreeing to be Derek’s fake date for Cora’s wedding is the stupidest thing he’s ever done, but it’s a little difficult to say no to the man he’s been in love with for seven years.

I Just Want You For My Own (More Than You Could Ever Know) by yodasyoyo | 16K

“What is with that sweater, dude?”

Derek ducks his head to look at it, abashed. “Uh- Mrs Hernandez knitted it for me. It’s an early Christmas gift.” He smooths it down self-consciously.

Stiles cocks an eyebrow.

“What? She’s my neighbor and sometimes I-” Derek trails off. Stiles’ other eyebrow rises to join the first, and Derek sighs. “Sometimes I help her carry her groceries."

Of course he does. One day maybe Stiles will stop being in love with Derek Hale, but today is not that day.

Reason For Call by 74days | 58.9K | Explicit

Stiles has been working in his call-centre booth for nearly 5 years when he first hears the voice of the new IT guy. Surely anyone who sounds like that has got to be H.O.T Stiles takes it upon himself to get to know him better. The only problem is, he's got no idea what he looks like...

Stay in Your Lane by mikkimouse | 5.3K

Stiles works at a bowling alley and has a crush on Derek Hale, a frequent customer. He deals with this in the most mature way possible: by giving Derek dumb nicknames for every game he plays.

Pick Me Up by Omni | 4.7K

Growing more and more confident as he grew older, Stiles started the ridiculous pick-up lines and joke-flirting with Derek back when he was still in college. After college, it just got worse. Not that Derek minds. Sometimes Derek will play along, because Stiles is funny, and those bad pun innuendo pick-up lines are ridiculously hilarious, and he likes it when Stiles tosses his head back and laughs like that. He loves it when he’s able to catch Stiles off guard with his own brand of humor, loves the way his eyes light up as he smiles at Derek. Pretty much, he’s just kind of hopelessly gone on Stiles, but doesn’t fully realize it.

Then one day…he does.

Coming Home by sheafrotherdon | 9.9K | Explicit

When Stiles comes home from college for Thanksgiving break, the last thing he expects to develop is a sudden, overwhelming attraction to Derek Hale.

Where to Search for Snow by suburbanmotel | 8.9K | Mature

Stiles and his Gigantic Repressed Feelings accidentally affect the weather. A lot. Like. A lot.

A Blossoming Romance by Trelkez | 7.5K

Stiles will just have to try harder next time. No one can ignore him forever.

You've Got Notes by the_gramophone | 14.8K | Mature

Stiles Stilinski has wanted star basketball player Derek Hale forever, but what are the odds of that ever happening? A love story of letters, prom, and the healing power of milkshakes.

Smooth Like Your Face by Cobrilee | 2.5K

Derek is so used to Stiles hitting on him with horribly cheesy pick-up lines, he doesn't realize that maybe Stiles means them. Luckily, Boyd does.

Worth the Wait by Dexterous_Sinistrous | 13.3K | Explicit

Stiles always had a thing for Derek, but then again, so did everyone else. Stiles just wanted to be seen as different, which was why he waited.

But maybe he waited a little too long.

Binomial Coefficients by DevilDoll | 20.7K

In which brainy freshman Stiles Stilinski wants star quarterback Derek Hale to join the math team, AKA math nerds in love.

Breaking Bad Habits by Inell | 3.7K

Stiles has a bad habit of falling for unattainable higher beings that would never so much as look at a mere mortal like him. The latest focus of his unrequited affection is his personal trainer, Derek Hale, who is a Greek God come to life.

Go For The Gold (And A Few Other Things) by SpiritsFlame | 14.7K | Mature

Stiles came to the Olympics with one goal- get a gold medal. By Opening Ceremonies, he has two goals. Win a gold medal, and sleep with Derek Hale. Unfortunately for him, those two goals are equally difficult.

[coraniaid]

top 5 mathematical identities :)

I wish Tumblr supported LaTeX.

OK, not trying to overthink this too much.

There are lots of fun identities involving binomial coefficients (or their q-analogs), or related integer sequences like the Catalan numbers and Motzkin numbers. But I think I have to go with the Chu-Vandermonde identity: who doesn't like a good convolution formula?

Of all the trigonometric identities I've ever had to learn, this is certainly not the most useful in practice or the hardest to prove or, arguably. the most inherently interesting either. I think the half-angle formula for tan is surprisingly pretty though.

Let S be a finite set of real numbers. The maximum-minimums identity relates the largest element of this set to the smallest elements of every (non-empty) subset of S.

or, more concretely,

Perhaps not classically beautiful, but certainly enormously useful, the Sherman-Morrisson-Woodbury identity in linear algebra gives a formula for computing the inverse of a rank-k update of an invertible matrix by doing rank-k updates of the inverse of that original matrix. It's valid whenever the matrices are suitably conformable and when both the required inverses exist.

I feel like I have to include something due to Euler here, but -- rather than one of the famous ones involving π or anything to do with topology -- I'll go with some analytic number theory. The pentagonal number theorem gives a series expansion of the Euler function, valid for any complex x in the unit circle.

or, expanding both sides,

That takes us into the world of q-series and we can generalize it further to get Jacobi's triple product formula or various identities due to Ramanujan or MacDonald's identities for affine root systems or other increasingly exotic and strange things ... but this identity is the prototype for all of them.

[boycritter]

hey girl is that pascal’s triangle in your pocket or is your penis just a triangular array of binomial coefficients that’s useful in probability theory and combinatorics

[justanothersterekficgirl]

Binomial Coefficients by DevilDoll 

Teen | 20k | 1/1

In which brainy freshman Stiles Stilinski wants star quarterback Derek Hale to join the math team, AKA math nerds in love.

[jessecmckeown]

Whence Fabulous Faulhaber?

should I promise not to make this a habit?

Dear Mr. Haran, I'm grateful to you and correspondent Eischen (and Conway) for putting the name of Faulhaber on a calculation which heretofore I'd only known quoted, without attribution, by Heinrich Dörrie in Triumf der Mathematik---(which I've only read in translation). However, I'm most frustrated that neither Dörrie nor Eischen give any satisfying motivation for why the postulate should work.

For bystanders still catching up, this postulate is that if one defines a sequence of numbers $B_k$ "by expanding" $$(B-1)^{k+1} = B^{k+1}$$ and transcribing exponents to subscripts... one finds that the differences $$ (n + B)^{k+1} - B^{k+1} $$ similarly treated are equal to cumulative power sums, $$(k+1) \sum_{j \leq n} j^k$$

So the calculation is doable. My Beef is Dilemmimorphic: Either the notational abuse of $(n + B)^k$ suggests that $B$ should be Some Kind Of Linear Operator, in which case what is it? Or else there's an Amazing Coincidence being Overlooked!

It's a comparative Triviality that the power sums $\sum_1^N n^k$ should be polynomials of $N$, and that the leading term be $\frac{1}{k+1} N^{k+1}$ , so indeed it is perfectly reasonable to consider coefficients $B_{k,j}$ defined by $$ \sum_1^N n^k = \frac{1}{k+1} \sum \binom{k+1}{j} B_{k,j} N^{k+1-j} $$ BUT WHY SHOULD WE ASSUME that in fact $B_{k,j}$ depends only on $j$? That's STAGE MAGIC, and the fact that indeed it somehow works does not explain "where it comes from" (Eischen's favourite phrase on the matter).

So, in my customary way of starting with the actual problem and throwing at it what seems to me the minimum of thought, let's first explicate that "comparative triviality": the sequence of polynomials $p_k(j) = \binom{j+k}{j}$ are integral generators for the Integral-valued polynomials, and are recursively definable as iterated cumulative sums of the constant polynomial $p_0 \equiv 1$: $$\binom{j+k+1}{j} = \binom{j+k}{j} + \binom{j+k}{j-1}$$. Hence, cumulative sums of any polynomial, written in the binomial basis, can be obtained just by incrementing: $$\sum_{j=1}^N \sum a_n p_n(j) = \sum a_n p_{n+1}(N)$$

Next, cumulative sums are themselves defined by induction: $"\sum_{j=1}^0" P(j) = 0$ and $\sum_{j=1}^{N+1} P(j) = P(N+1) + \sum_{j=1}^N P(j)$, or said differently, by the Difference equation $$ SP(N+1) - SP(N) = P(N+1).$$ In other words we are trying to solve the Difference Equations $$ S_k(N) - S_k(N-1) = N^k,$$ but in the basis of Monomials $N^j$ instead of Binomials $p_j(N)$.

The binomial theorem, $$ (x+y)^k = \sum \binom{k}{j} x^{k-j} y^j $$ makes the Taylor-MacLaurin formula a Theorem for polynomials $$ (x+y)^k = \sum y^j \frac{1}{j!} \frac{d^j}{dx^j} x^k $$ which is fruitfully abbreviated $$ P(x+y) = e^{y\\, d/dx} P(x) $$ the Backwards Difference, then, is similarly $$ P(x) - P(x-1) = (1 - e^{- d/dx}) P(x) $$

Shall we say, The kernel of the Backward Difference is reasonably well understood? The differential operator is the retract of the Integral operator $\int_0$, so the Taylor-MacLaurin formula provides us also a section for the Forward Difference operator, $$ 1-e^{-x} = \frac{d}{dx} + A\frac{d^2}{dx^2} $$ where, for now, the main point is that the unbounded-degree differential operator $A$ commutes with $d/dx$, so that, for example $$ (1 - e^{-d/dx}) \left(\int_0 \sim dx - A + A^2 \frac{d}{dx} - A^3\frac{d^2}{dx^2} + - \cdots \right) P(x) = P(x)$$

Of course, there are various paths to the power series, other than via expansion of the powers of $A$, but there is a (Laurent) power series $$ \frac{1}{1-e^{-t}} = \frac{1}{2}\coth(\frac{t}{2})+\frac{1}{2} = \frac{1}{t} + \sum \frac{B_j}{j!} t^{j-1} $$ where $B_j$ are the faBulous Bernoulli numbers.

In any case, applied to simple powers, $$ \left( \int_0 \sim dx + \frac{1}{2} + \sum_{j=2}^{\infty} \frac{B_j}{j!} \frac{d^{j-1}}{dx^{j-1}} \right) x^k = \frac{1}{k+1} x^{k+1} + \sum_{j=1}^{k} \frac{k!}{j!(k-j+1)!} x^{k-j+1} B_j \\\\ {} = \frac{1}{k+1} \sum_{j=0}^{k} \binom{k+1}{j} B_j x^{k+1-j} $$ Finally, the power sum polynomials $S_k$ vanish both at zero (formally an empty sum) and at $-1$ (since $S_k(0) - S_k(-1) = 0^k$), so that in particular, $$ \sum_{j=0}^k \binom{k+1}{j} B_j (-1)^{k-j} = 0$$ THAT'S WHERE THIS IS COMING FROM.

[buckttommy]

hi! jumping on what the anon's started, what's your favourite or your comfort Sterek fic? It's been a long time since I've read a good one for them...

hello! i wanted to wait until i was finished school for the day to answer this hehe. okay, so, my favorite fic and my comfort fic are two different things, but here are some fun ones!

favorite fic: play crack the sky | fixer upper

comfort fic: piece of mind

ashes, ashes

no rest for the wicked

binomial coefficients

by any other name

icarus come back down

in the practice of my calling

some of these i haven't read in a while so i don't really remember them, but my faves and my comfort fic of course i remember. i hope you enjoy !

[vitaecryptid]

🪐 ⇢ name three good things going on in your life right now

🍬 ⇢ post an unpopular opinion about a popular fandom character 🧩 ⇢ what will make you click away from a fanfiction immediately? 🔪 ⇢ what's the weirdest topic you researched for a writing project?

🪐: This one's hard cus I'm stressed dfgcnhkfgjnhg. I'd say... People taking my furniture off my hands is nice LOL. The thought of moving in with my best friends is keeping me going. And... The extremely positive reception of a fic I was really nervous to post cus it was in a weird format.

🍬: ...Oh boy. UH. Alhaitham isn't an uncaring asshole people just lack reading comprehension ✌️

🧩: The biggest one is people being ooc. Especially if someone overly feminizes a male character that doesn't act feminine in canon.

🔪: Easy. The Binomial Coefficient equations to calculate the odds of specific hands in poker.

[myprogrammingsolver]

Problem 1 – Pascal’s Triangle:

Pascal’s triangle is a geometric arrangement of sums that have interesting mathematical properties – most famously, the binomial coefficients. The rows of Pascal’s triangle are conventionally enumerated starting with row 0, and the numbers in each row are usually staggered relative to the numbers in the adjacent rows. A simple construction of the triangle proceeds in the following manner. On row…

[wiredfaculty0]

Essential Formulas and Theorems for 12th Class Mathematics Board Exams

The 12th class Mathematics board exams in India cover a wide range of topics. Here are some essential formulas and theorems that you should be familiar with for the exam:

Algebra:

Quadratic Formula: For a quadratic equation of the form ax^2 + bx + c = 0, the solutions can be found using the quadratic formula: x = (-b ± √(b^2 – 4ac)) / (2a)

Factorization Formulas:

(a + b)^2 = a^2 + 2ab + b^2

(a – b)^2 = a^2 – 2ab + b^2

a^2 – b^2 = (a + b)(a – b)

a^3 + b^3 = (a + b)(a^2 – ab + b^2)

a^3 – b^3 = (a – b)(a^2 + ab + b^2)

Binomial Theorem: The expansion of (a + b)^n can be found using the binomial theorem: (a + b)^n = C(n, 0) * a^n * b^0 + C(n, 1) * a^(n-1) * b^1 + … + C(n, n) * a^0 * b^n where C(n, r) denotes the binomial coefficient.

Read our blog for more information :-

[math-monthly]

MULTIPLYING BINOMIAL EXPRESSIONS USING FOIL

FOIL is an effective and easy method of multiplying binomial expressions.

To use FOIL, multiply the first two numbers in the brackets (they should both be the unknown value and it's coefficient.) Then, multiply the outside numbers, and then the inside, and then the last numbers.

Let's use FOIL in the example below:

First : 2x * 6x = 12x² Outside : 2x * 7 = 14x Inside : 4 * 6x = 24x Last : 4 * 7 = 28

Add all the values together, combining like terms.

12x²+ 14x + 24x + 28 = 12x² + 38x + 28

[programmingsolver]

Problem 1 – Pascal’s Triangle:

Pascal’s triangle is a geometric arrangement of sums that have interesting mathematical properties – most famously, the binomial coefficients.   The rows of Pascal’s triangle are conventionally enumerated starting with row 0, and the numbers in each row are usually staggered relative to the numbers in the adjacent rows. A simple construction of the triangle proceeds in the following manner. On…

[tutoroot]

What are the types of polynomials?

Polynomials are algebraic expressions with variables and coefficients in them. Polynomials are a type of mathematical dialect. They are used to express numbers in almost every field of mathematics and play an essential role in others, such as calculus.

A polynomial is a sort of algebraic statement in which all variable exponents are whole numbers. Variable exponents in any polynomial must be non-negative integers. A polynomial is made up of constants and variables, where we cannot divide polynomials by variables. 

Polynomials are classified into the following groups based on the number of terms: 

Monomial

A monomial expression has only one term. The single term in an expression must be non-zero for it to be a monomial.

Binomial

A binomial expression has two terms. A binomial is the sum or difference of two or more monomials.

Trinomial

A trinomial expression has three terms.

We utilize the long division approach when a polynomials contains more than one term. The steps are as follows. 

In decreasing sequence, write the polynomial. Examine the greatest power and divide the phrases by it. As the division symbol, use the solution from step 2. Subtract it from the following word and bring it down. Steps 2–4 should be repeated until there are no more words to carry down. It is important to note that the final answer, including the remainder, will be in fraction form (last subtract term). 

Enroll in Tutoroot to receive one-on-one Math’s online interactive classes on a variety of topics. Register today to have access to a plethora of courses covering various subjects in a more effective manner.