Talk:Binomial coefficient

message: Made an edit from (n-(k-1)) to (n-(k+1)) which should be the correct falling factorial. If I'm wrong revert me please. 
2601:640:4100:4C70:81FA:2C7B:8F13:A0D4 (talk) 22:17, 5 September 2019 (UTC)
replies: text: You're wrong so I have reverted you. Maybe you missed that n-(k-1) = n-k+1. PrimeHunter (talk) 23:29, 5 September 2019 (UTC)
replies: text:  Doh, completely spaced. It just felt weird at first glance. Thanks. 2601:640:4100:4C70:81FA:2C7B:8F13:A0D4 (talk) 00:42, 6 September 2019 (UTC)
thread_title: Multiplicative formula

message: 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)
replies: text: 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)

text: 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)
replies: text: 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)

text: 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)
replies: text:  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)
replies: text:  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)
replies: text:  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)
replies: text:  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)
replies: text:  Good point. --JBL (talk) 22:06, 13 August 2012 (UTC)
thread_title: Binomial Coefficient definition should precede discussion of Binomial Theorem and Pascal's Triangle

message: How is  \binom nk  usually pronounced?
Shouldn't this be in the article? 84.29.139.151 (talk) 09:43, 9 June 2014 (UTC)
replies: text: Try reading the second paragraph of the lead section ;).  JBL (talk) 13:19, 9 June 2014 (UTC)
replies: text:  Since this part of the text has changed since the above comment was written 7 years ago, I'll quote the relevant sentence from that point in time: \tbinom nk is often read as "n choose k". 194.39.218.10 (talk) 13:45, 2 July 2021 (UTC)
replies: text:  I've added this sentence back to the main article. 194.39.218.10 (talk) 13:58, 2 July 2021 (UTC)
replies: text:  You are mistaken that it has changed -- I mean, perhaps the location is different now than it was, but it's still in the lead, following "The binomial coefficients occur in many areas of mathematics, and especially in combinatorics."  So I have reverted your redundant addition. --JBL (talk) 14:10, 2 July 2021 (UTC)
thread_title: Pronunciation?

message: 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)
thread_title:  Another identity 

message: 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)
replies: text: 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)
thread_title:  Equation numbering 

message: 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)
replies: text: I think it's implicit in the section that  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)
replies: text:  I'd rather say that implicitly k-(n/2) = o(n); worth to mention this. 192.54.190.20 (talk) 16:37, 3 July 2015 (UTC)
thread_title:  Approximation 

message: 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)
replies: text: , 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)
thread_title:  Omar Khayyam triangle and Yang Hui triangle 

message: I removed the following:
The apparent reference (which I also removed, not given inline, but probably should have been) is here
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)
replies: text: 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)
replies: text: 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)
thread_title:  Result of Singmaster 

message: concludes with an interesting result (namely, that primality is equivalent to dividing all the intermediate binomial coefficients), but those statements look suspiciously like WP:OR.  (Note, for example, that the paragraph is unsourced and includes a proof.)  I couldn't find a source with a cursory Google search; does anyone else recognize the result as being published somewhere?  128.135.98.14 (talk) 18:03, 15 February 2019 (UTC)
replies: text: There's a somewhat similar (but different) primality test based on looking at non-horizontal lines through Pascal's triangle in . This test itself is called "well-known" and is the starting point for the main results in a more recent paper,  —David Eppstein (talk) 22:32, 15 February 2019 (UTC)
thread_title:  WP:OR? 

message: I edited the first equation under "generating functions" => "Ordinary generating functions" 
Previously read 
: \sum_{k=0}^\infty {n\choose k} x^k = (1+x)^n.
now reads
: \sum_{k=0}^n{n\choose k} x^k = (1+x)^n.
Summing k above n is mildly confusing at best. — Preceding unsigned comment added by Affine.function (talk • contribs) 
: It would not be confusing to you if you read the article, where this is discussed at least twice: Binomial_coefficient#Generalization_and_connection_to_the_binomial_series and Binomial_coefficient#Binomial_coefficients_as_polynomials. --JBL (talk) 12:31, 15 December 2020 (UTC)
thread_title:  edit to "generating functions" 

message: There are some results.
My results are:
k=1, n=2^82589933–1=148894445742...325217902591 (24862048 digits; elipsis omits 24862024 digits) (see largest known prime);
k=2, n=2618163402417 * 2^1290001 - 1 (388342 digits) (see Cunningham chain).
Can you extend my results for k>2? — Preceding unsigned comment added by 109.106.227.160 (talk • contribs) 18:01, 25 August 2021 (UTC)
replies: text: Unless you are basing this on reliable publications rather than your own original research, it is off-topic here. This page is for discussions on improvements to the article, which can only be based on sources. —David Eppstein (talk) 18:27, 25 August 2021 (UTC)

text: If you want people to assist with calculations, this is not the place. Try the WP:HELPDESK. —David Eppstein (talk) 16:50, 27 August 2021 (UTC)
thread_title: Largest currently known positive integer n such that binomial(n,k) has exactly k prime factors, counted with multiplicity

message: * Specific text to be added:
If n is a positive integer  and p is a prime number, then  where \nu_p(n) is the -adic order or -adic valuation of an integer , being the exponent of the highest power of the prime number  that divides . Equation () follows from a more elementary fact. If j is a positive integer,
 \binom {n-1}{p^j - 1} \equiv \begin{cases} 1 \mod p & \text{if }\ p^j | n \\ 0 \mod p & \text{if }\ p^j \nmid n. \end{cases}
* Reason for the change: I humbly think this result is a valid addition to the body of knowledge on this topic. Additionally, I believe that there is no review article in which the result I demonstrated is stated.
* References supporting change: http://math.colgate.edu/~integers/w61/w61.pdf
Dario T. de Castro (talk) 10:32, 1 September 2022 (UTC)
replies: text: Hey there, which article do you want to add this to? Thanks, --Ferien (talk) 19:05, 1 September 2022 (UTC)
replies: text: Dear Ferien, the addition was intended for the article on 'Binomial Coefficients', in the 'Congruences' section.Dario T. de Castro (talk) 23:59, 1 September 2022 (UTC)

text: See my comment below for the other requested addition. My reaction here is the same. —David Eppstein (talk) 13:52, 26 September 2022 (UTC)
thread_title:  Some proposed changes in the article on Binomial Coefficients - section Congruences 

message: Hi, in french the formula k{n \choose k} =n{n-1 \choose {k-1}}  is named "pawn formula" : https://fr.wikipedia.org/wiki/Formule_du_pion.  Is there a name in English ? Robert FERREOL (talk) 11:39, 25 November 2022 (UTC)
replies: text: FWIW, Concrete Mathematics calls the identity (or a slightly rearranged form of it) "absorption/extraction".  A casual search finds some other uses of the term absorption, although I'm not certain it is widespread; I didn't find the term extraction. Russ Woodroofe (talk) 11:50, 25 November 2022 (UTC)
replies: text: Thank you. I added this reference on  https://fr.wikipedia.org/wiki/Formule_du_pion. Robert FERREOL (talk) 09:31, 27 November 2022 (UTC)
replies: text: I have never felt that this identity had a standard English name. --JBL (talk) 17:38, 27 November 2022 (UTC)
thread_title:  Name for an identity 

message: Can the section on programming languages be blanked? It adds nothing at all to the article; it reads like a homework assignment for high-school seniors. There are interesting things to say about algorithmic identities for binomials, but those algorithms customarily appear first in basic hypergeometric series (and date back to the 19th century, so pre-computer algorithms. I think Kummer gave some of the first ones.) So, yes, there could be a section on those algorithms, but this is not it. 67.198.37.16 (talk) 22:45, 5 January 2024 (UTC)
replies: text: I was bold and just blanked. Looking at the page history, this section was an accretion of original research that was added over the years by anon authors making a case for their favorite programming languages. 67.198.37.16 (talk) 22:55, 5 January 2024 (UTC)
thread_title:  Blank section on programming languages? 

message: The redirect NCk has been listed at redirects for discussion to determine whether its use and function meets the redirect guidelines. Readers of this page are welcome to comment on this redirect at  until a consensus is reached.  Shhhnotsoloud (talk) 18:47, 6 February 2024 (UTC)
thread_title:  "[[:NCk]]" listed at [[Wikipedia:Redirects for discussion|Redirects for discussion]]