# $\binom{n}{r}$ versus $^n\mathrm{C}_r$ : which notation is more used?

I know that the notation $\binom{n}{r}$ is more standard to use since we have a $\LaTeX$ command for it while there is no such thing for $^n\mathrm{C}_r$.

Now, I'm wondering which notation do people use in general when writing by hand. Also, I researched a bit about which countries use which notation mostly but it was futile since I didn't get effective results.

I'd just like to know which notation is more used in general and whether the $^n\mathrm{C}_r$ is used anywhere outside India (I'm Indian, so I'd like to know). Also, I'd like if someone can provide me with some sort of a "poll" of which notation is more used in which countries.

P.s - I have tagged it as a soft question, so I hope this doesn't get closed as "too broad" or by any similar reason. Thanks!

• There is also a discussion about the notation here. – Dietrich Burde May 12 '15 at 15:16
• I'll answer as a comment because what I are going to say is not based on data, but in my personal experience. I'd say that $\binom nr$ is used at most contexts, but $^n\mathrm C_r$ it is usually seen only in combinatorial contexts. – ajotatxe May 12 '15 at 15:17
• It's rare to see ${}^nC_r$ or $C_{n, r}$ outside elementary combinatorics (and you won't see it anywhere at all outside combinatorics), though I think I have seen some more advanced combinatorics papers that use $C(n, r)$. – anomaly May 12 '15 at 15:23
• Not an answer, but $C_n^r$ and $C_r^n$ are both in use. – André Nicolas May 12 '15 at 15:34
• I've seen all of the notations discussed in this thread, but certainly the most commonly used one was $\binom{n}{r}$. – izœc May 13 '15 at 9:23

Update I've added two more notation (seen on this site), which might be relevant. The data change afterwards, mostly in the exact matches in $\LaTeX$Search.

As others stated, I would go with $\binom{n}{r}$(probably never used different notation, although that does not mean much, since I am in math for only few years).

Although for example wiki (search for "binom") gives more options.

As to support my (as well as others) choices I can give you some numbers obtained by searching the terms here ($\LaTeX$Search) and here ($\LaTeX$SpeedSearch). Both hyperlinks link to the page claiming

The Springer LaTeX search lets you search through over 8,223,138 LaTeX code snippets to find the equation you need.

Claim

I did not check all the results, whether they really mean the binomial coefficient, so bear that in mind.

\begin{array}{|c|c|c|c|c|} \hline \text{Term} & \text{in }\TeX & \text{Search}^* & \text{SpeedSearch} \\ \hline \binom{n}{k} &\verb+\binom{n}{k}+ & 10/1464 & 323 \\ \hline \binom{n}{r} &\verb+\binom{n}{r}+ & 1/815 & 47 \\ \hline \binom{n}{2} &\verb+\binom{n}{2}+ & 13/0 & 166 \\ \hline \binom{n+1}{k} & \verb-\binom{n+1}{k}- & 0/14 & 14 \\ \hline ^nC_r & \verb+^nCr+ & 3/301 & 2 \\ \hline ^nC_k & \verb+^nC_k+ & 13/830 & 3\\ \hline ^nC_2 & \verb+^nC_2+ & 1/619 & 5 \\ \hline C_r^n & \verb+Cr^n+ & 1/189 & 74 \\ \hline C_k^n & \verb+C_k^n+ & 1/316 & 13\\ \hline C_2^n & \verb+C_2^n+ & 6/690 & 8\\ \hline C_n^r & \verb+C_n^r+ & 3/229 & 7 \\ \hline C_n^k & \verb+C_n^k+ & 11/287 & 37 \\ \hline C_n^2 & \verb+C_n^2+ & 39/1471 & 78 \\ \hline C(n,r) &\verb+C(n,r)+ & 0/1317 & 9\\ \hline C(n,k) &\verb+C(n,k)+ & 2/1432 & 32 \\ \hline C(n,2) &\verb+C(n,2)+ & 0/1771 & 4\\ \hline \end{array}

$^*$ The first number responds to the exact result appearance, the second to the similar one.

Conclusion

Personally I would not really consider the similar results for $\LaTeX$Search. Of course, this statement should be supported by going through the results (and obtaining, that most results are not binomial coefficients). Excluding those (and the $\binom{n+1}{k}$ row) out we get

\begin{array}{|c|c|c|} \hline \text{Type of notation} & \sum Search & \sum SpeedSearch \\ \hline \text{Notation } \binom{n}{x} & 24 & 536 \\ \hline \text{Different notation} & 80 & 272\\ \hline \end{array}

Note I did this mostly for fun. I think these sites have better usage then for some notational statistics.

• This is exactly what I needed! Thanks! It'd have been much better if we could compare usage of notations between countries. – notationer May 12 '15 at 17:10
• You might want to add that also n \choose k and matrix commands are used for the first notation. – wythagoras May 13 '15 at 14:28

I've seen ${}^n C_r$, but almost vanishingly rarely. The notation $n \choose r$ is easily the most common in my experience, but one sees $C(n, r)$ often too.

• In my country, at least in highschool, no one uses the bracket notation. I bet most students wouldn't know what it means. Also, we put both the $n$ and $r$ at the right side. It'd look weird to put one on each side, at least to us. – Hasan Saad May 12 '15 at 15:27
• @HasanSaad Interesting, what is your home country? (FWIW, I was educated in the U.S.) – Travis Willse May 12 '15 at 15:33
• Lebanon, Middle East. – Hasan Saad May 12 '15 at 15:40

Certainly $n \choose r$, i have never seen ${}^n C_r$. Other than that i see $nCr$ as this is the standard input on a calculator

I think the notation $n \choose r$ is most common. I've never seen $^n C_r$, actually, but in cases that it must be written without LaTeX, I often see nCr. Another notation I sometimes see are $C_r^n$ or $C(n,r)$.