# Talk:Binomial coefficient

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## Binomial Coefficient definition should precede discussion of Binomial Theorem and Pascal's Triangle

The definition of the binomial coefficient in terms of what it means numerically should precede discussion of binomial theorem. Reader quite possibly may only want to know what it means and have no interest in binomial theorem or Pascal Triangle.RHB100 (talk) 21:31, 11 August 2012 (UTC)

This is not a difficult article to navigate: a person interested in formulas for binomial coefficients need only look at the table of contents for the well-labeled sections with the word "formula" in their titles. Please see Wikipedia:Manual of Style/Lead section to read about what a lead section is supposed to accomplish; notably, "The lead should be able to stand alone as a concise overview. It should define the topic, establish context, explain why the topic is notable, and summarize the most important points." Your sentence is detrimental to almost all of these goals, and distracts emphasis from the important definitions to less-important formulas. --JBL (talk) 02:24, 11 August 2012 (UTC)

The article at present is either deliberately designed to confuse readers or it is so poorly written that it confuses the reader as much as if it had been deliberate. The statement above implying that the definition of the binomial coefficient , ${\displaystyle {\binom {n}{k}}={\frac {n!}{k!\,(n-k)!}}\quad {\mbox{for }}\ 0\leq k\leq n}$, where the notation, ${\displaystyle n!}$, called n factorial is defined by ${\displaystyle n!=n(n-1)(n-2)...1}$ with ${\displaystyle 0!=1}$, is of less importance than Pascal's triangle and such is a very ignorant and stupid statement. RHB100 (talk) 21:31, 11 August 2012 (UTC)

Go have a read of WP:Civility and perhaps we can talk. It seems to me possible that a formula (or a pointer to the section on formulas) could be worked into the introduction somewhere, but you're very much going about it the wrong way. --JBL (talk) 18:11, 13 August 2012 (UTC)
I fully agree with that —specially about the civility issue. The factorial expression could i.m.o. be easily incorporated in the lead — not the way RHB100 has in mind though, as that would indeed be 100% orthogonal to Wikipedia:Manual of Style/Lead section. I have made an attempt at doing so. I think this can work. - DVdm (talk) 18:43, 13 August 2012 (UTC)

OK, I think it is somewhat better now. However, I think it should be kept in mind that some people may not be familiar with the factorial operator. Therefore I think a definition of the factorial operator for the simplest case of positive integers and zero before getting into the more advanced part of the article would be desirable. RHB100 (talk) 19:15, 13 August 2012 (UTC)

Mentioning factorials would i.m.o. clutter up the lead. The reader is explicitly pointed to factorials in all their glory in the section Binomial coefficient#Factorial formula. Remember the guidelines about the lead: the keywords to go with are concise overview and summarize. - DVdm (talk) 20:00, 13 August 2012 (UTC)
I think the current phrasing (while obviously better than the original) is somewhat cluttered -- that sentence is now doing an awful lot of work. Better would be to make it its own sentence (a little later in the paragraph). In fact, though, I think it would be even better to replace the formula with a sentence like the following: "There are several simple formulas (both recursive and explicit) for computing binomial coefficients -- see the section below for details." This serves the purpose of mentioning and pointing to the formulas without cluttering the lead or detracting from the more important definitions (namely, that these are the coefficients that appear in the binomial theorem, and their combinatorial interpretation). --JBL (talk) 20:09, 13 August 2012 (UTC)
Hm, I wouldn't opt for a see-below-like-kind-of-easteregg like you suggested. But yes, I see your point. Indeed, the sentence is doing too much work now, so I have moved the calculation into its own sentence — let's have the work done by two sentences, right? There are indeed several simple formulas, but I dare assume, so to speak, that we can all agree that the factorial expression is the most common and most notable... so how about this? - DVdm (talk) 21:29, 13 August 2012 (UTC)
Would you object to replacing "Under suitable circumstances" with "When n and k are nonnegative integers"? Should there be a remark about the fact that they can be defined more generally? Or is this too much for the lead? Edit: ok, the remark about generalizing comes later, is fine there, and shouldn't be repeated. The first question stands. --JBL (talk) 21:43, 13 August 2012 (UTC)
Re the first question: I wouldn't mind —provided n-k is required to be a nonnegint as well, which would add slightly more clutter again— but I don't think we really need that in this lead. - DVdm (talk) 22:05, 13 August 2012 (UTC)
Good point. --JBL (talk) 22:06, 13 August 2012 (UTC)

(In english please. It's hard to know what is just jibber jabber and what is solid useful math language!)

## Pronunciation?

How is ${\displaystyle {\binom {n}{k}}}$ usually pronounced? Shouldn't this be in the article? 84.29.139.151 (talk) 09:43, 9 June 2014 (UTC)

Try reading the second paragraph of the lead section ;). JBL (talk) 13:19, 9 June 2014 (UTC)

## Another identity

You have (under 'Identities involving binomial coefficients') everything but the basis for the recursive formula that appears in 'Computing the value of binomial coefficients':

nCk = (n-1)C(k-1) + (n-1)Ck

when I copy/paste I get: \binom nk = \binom{n-1}{k-1} + \binom{n-1}k \quad \text{for all integers }n,k : 1\le k\le n-1

The second identity down is similar looking, but the recursive formula seems more direct.

It just seems like the 'recursive identity' should be included. 71.139.161.9 (talk) 05:12, 28 August 2014 (UTC)

## Equation numbering

Although I approve of equation numbering (as found in this article) and certainly use it in my own publications, I am finding it a hinderance to editing this article. I am considering numbering an equation early in the article which doesn't have a number now. This will require a wholesale renumbering of all the equations below it and their reference tags. Unwillingness to do this may explain why a blatant but minor error has not been corrected for years. When I do this I will remove all equation numbers that are not referred to (I know that many authors put them in, "just in case", they need to refer to the equations, or for the possibility that they might want to do this in the future ... but it creates more work for editors) and I will move to a numbering within section scheme (equations numbered 2.1, 2.2, ..., 2.k in section 2 and 3.1, 3.2 ... in section 3, and this can be extended to subsections as well). This will mean that fewer changes are needed when additions and deletions are made to numbered equations. Comments? Bill Cherowitzo (talk) 06:55, 4 December 2014 (UTC)

I certainly approve of not numbering equations that we don't refer to. And in some cases, like (3), there is an actual name we can refer to ("Pascal's relation") that might make the text clearer as well as avoiding the need for numbers. As for the section-based numbering, I wonder if it really will make for fewer changes? Then we'd have to change all the numbers every time we change the section heading organization rather than every time we add or remove a numbered equation; both seem likely to be infrequent, but still often enough to be annoying. I just added my own first WP "numbered" equation a couple of days ago (in Euclid–Euler theorem‎) but in that case there was only one so I just used (*). —David Eppstein (talk) 07:07, 4 December 2014 (UTC)

I realized the problem with section reorganization shortly after I wrote the above. Perhaps using a letter scheme (A.1, A.2, etc. for section 3; B.1, B.2, etc. for section 5 (with no numbered equations in section 4)) and not going any deeper than main section levels would be about the best one could hope for given the fluid nature of our articles. I think that stars and names are to be preferred whenever they make sense. Bill Cherowitzo (talk) 18:03, 4 December 2014 (UTC)

## Approximation

Hi, I explicitely checked the formula in the subsection APPROXIMATION, with Mathematica 10.0 and it is clearly wrong since the ratio of the two members goes to zero instead of 1. Could please someone tell me where to find the derivation of such a formula? — Preceding unsigned comment added by 163.1.241.224 (talk) 18:43, 11 May 2015 (UTC)

I think it's implicit in the section that ${\displaystyle k=\Theta (n)}$, though of course this isn't stated explicitly. (And why is this section not part of the preceding section, anyhow? Sigh.) --JBL (talk) 21:10, 11 May 2015 (UTC)
I'd rather say that implicitly ${\displaystyle k-(n/2)=o(n)}$; worth to mention this. 192.54.190.20 (talk) 16:37, 3 July 2015 (UTC)

## Omar Khayyam triangle and Yang Hui triangle

User:Joel B. Lewis, you removed information about Yang Hui's triangle, with the argument this is a redirection, this is not the case, this was as now a link to the author of the triangle, I also added a link to Omar Khayyam, as it also created this triangle in Iran. It is interesting to have history of this triangle. I don't understand why you want to remove it.Popolon (talk) 01:36, 18 October 2016 (UTC)

Popolon, the purpose of the lead section of an article is to summarize the body of the article. If you would like to include historical information, you should put it in the section titled "History," along with an appropriate source. --JBL (talk) 02:11, 18 October 2016 (UTC)

## Result of Singmaster

I removed the following:

A somewhat surprising result by David Singmaster (1974) is that any integer divides almost all binomial coefficients. More precisely, fix an integer d and let f(N) denote the number of binomial coefficients ${\displaystyle {\tbinom {n}{k}}}$ with n < N such that d divides ${\displaystyle {\tbinom {n}{k}}}$. Then
${\displaystyle \lim _{N\to \infty }{\frac {f(N)}{N(N+1)/2}}=1.}$

Since the number of binomial coefficients ${\displaystyle {\tbinom {n}{k}}}$ with n < N is N(N + 1) / 2, this implies that the density of binomial coefficients divisible by d goes to 1.

The apparent reference (which I also removed, not given inline, but probably should have been) is here[1]

The statement (as I'm interpreting it) appears to be false. For example, for d = 2, take N = 2M, and then f(N) is exactly 3M. So the expression inside the limit can be made arbitrarily small, certainly not something approaching 1.

Unfortunately, I can only view the first page of the Singmaster article, so if anyone else has access to it and can correct the statement, that would be great. Or maybe I'm just being dense, in which case, please revert me with extreme prejudice. --Deacon Vorbis (talk) 22:10, 2 May 2017 (UTC)

References

1. ^ Singmaster, David (1974). "Notes on binomial coefficients. III. Any integer divides almost all binomial coefficients". Journal of the London Mathematical Society. 8 (3): 555–560. doi:10.1112/jlms/s2-8.3.555.
The number of even binomial coefficients up to 2M is certainly not 3M. It is the number of odd binomial coefficients that equals 3M. —David Eppstein (talk) 22:17, 2 May 2017 (UTC)
Yeah, sorry; I had a bad feeling it had to be something stupid on my part. Thanks. --Deacon Vorbis (talk) 00:16, 3 May 2017 (UTC)