Combination notation vs. Binomial Coefficient Formula

Question

I'm studying probability and statistics and had a question regarding notation.

I noticed that combinations and the binomial coefficient are essentially the same thing, that is:

$$\binom{n}{k}\ =\ _nC_k\ =\ \frac{n!}{(n-k)!k!}$$

But I was wondering, is there a particular difference between the two that people should be aware of? For example, are there certain use cases where one is preferred over the other?

Thank you.

Answer 1

All three expressions mean when dealing with combinations the same. But there are some aspects which should be considered.

We often find the binomial coefficients $\binom{n}{k}$ resp. $_nC_k$ defined by factorials. \begin{align*} \binom{n}{k}:=\frac{n!}{k!(n-k)!}\qquad\qquad\text{resp.}\qquad\qquad _nC_k:=\frac{n!}{k!(n-k)!} \end{align*} From this point of view the factorials $n!$ can be seen as basic building blocks for the shorthand notations $\binom{n}{k}$ and $_nC_k$. Since using factorials is more fundamental than the other two representations I will consider in the following only $\binom{n}{k}$ and $_nC_k$.

Historical aspects:

• C. Jordan writes in his classic Calculus of Finite Differences (1939)

  • ($\mathrm{\S}$ 22): Since the binomial coefficient is without doubt the most important function of the Calculus of Finite Differences it was necessary to adopt some brief notation for this function. We accepted above the notation of J. L. Raabe [Journal für reine und angewandte Mathematik 1851, Vol. 42, p. 350] which is most in use now, putting

  \begin{align*} \binom{x}{n}=\frac{x(x-1)(x-2)\cdots(x-n+1)}{1\cdot2\cdot3\cdots n} \end{align*}

  • ($\mathrm{\S}$ 22, footnote 18): Euler first used the notation $\left[\frac{x}{n}\right]$ in Acta Acad. Petrop.V, 1781 and then $\left(\frac{x}{n}\right)$ in Nova Acta Acad. Petrop. XV. 1799-1802. Raabe's notation $\binom{x}{n}$ is a slight modification of the second. It is used for instance in:
    • Bierens de Haan, Tables d'Int$\mathrm{\acute{e}}$grales d$\mathrm{\acute{e}}$finies, Leide, 1867
    • Hagen, Synopsis Vol. I. p. 57, Leipzig, 1891,
    • Pascal, Repertorium Vol. I, p. 47, Leipzig, 1910,
    • Encyclopädie der Math. Wissenschaften, 1898-1930,
    • L. M. Milne Thomson, Calculus of Finite Differences, 1933,
    • G. H. Hardy, Course of Pure Mathematics, p. 256, 1908

Euler's notation is also cited in

• A History of Mathematical Notations by F. Cajori, which also provides some information about the use of $_nC_k$-like notations in $\mathrm{\S}$ 451:

  • George Peacock (Treatise of Algebra, 1830) introduces $C_r$ for the combinations of $n$ things taken $r$ at a time.
  • Robert Potts (Elementary Algebra, 1880) begins his treatment by letting the number of combinations of $n$ different things taken $r$ at a time be denoted by $^nC_r$
  • W.A. Whitworth uses $C_r^n$ in Choice and Change (1886).
  • G. Chrystal writes $_nC_r$ in Algebra, Part II (1899).

Note the different variations $C_r, {^{n}C}_r, C^n_r$ of $_nC_r$-like notations.

Here is a selection of some classics from the 20th century.

Some classics:

• An Introduction to Combinatorial Analysis (1958) by J. Riordan

  • (3.1): ... and \begin{align*} C(n,r)=\frac{n(n-1)\cdots(n-r+1)}{r!}=\frac{n!}{r!(n-r)!}=\binom{n}{r} \end{align*} where the last symbol is that usual for binomial coefficients, that is, the coefficients in the expansion of $(a+b)^n$. ($C_r^n, C_n^r, _nC_r$ and $(n,r)$ are alternative notations; ...

• Combinatorial Identities (1968) by J. Riordan

  • (1.1): Perhaps the simplest combinatorial entities are the binomial coefficients, that is, the combinations, for example of $n$ things, $k$ at a time. They take their name from the generating function for combinations, which is a power of a binomial, namely \begin{align*} (1+x)^n=\sum_{k=0}^n\binom{n}{k}x^k \end{align*} where, of course, $\binom{n}{K}=C(n,k)=\frac{n!}{k!(n-k)!}$ is the usual notation for a binomial coefficient.

• An Introduction to Probability Theory and its Applications (1950) by W. Feller

  • (II.4): ... the number of subpopulations of size $r$ is therefore given by $(n)_r/r!$. Expressions of this kind are known as binomial coefficients and the standard notation for them is \begin{align*} \binom{n}{r}=\frac{(n)_r}{r!}=\frac{n(n-1)\cdots(n-r+1)}{1\cdot 2\cdots (r-1)\cdot r} \end{align*}

• Advanced Combinatorics (1974) by L. Comtet

  • (1.4): ... We will adopt the notation $\binom{n}{k}$, used almost in this form by Euler and fixed by Raabe, with the exclusion of all other notations, as this notation is used in the great majority of the present literature, and its use is even so still increasing. This symbol has all the qualities of a good notation: economical (no new letters introduced), expressive (it is very close in appearence to the explicit value $\frac{(n)_k}{k!}$, typical (no risk of being confused with others), and beautiful.

• Enumerative Combinatorics, Volume I (1986) by R. P. Stanley

  • (I.1.4): ... where $d_n=\sum_{i=0}^n\binom{n}{i}a_ib_{n-i}$, with $\binom{n}{i}=n!/i!(n-i)!$.

    and the author continues to use the notation $\binom{n}{k}$ without any more citations, indicating this notation being commonly used.

Typography, Readability:

• D. E. Knuth who gave us $\TeX$ is besides being a great mathematician an extraordinary expert in typography and mathematical writing. His Mathematical Typography (1979) gives us a glimpse of his deep thoughts about these issues. Another one being the report Mathematical Writing (1987) written together with T. Larrabee and P. M. Roberts.

  In $\TeX$ we use the command "$\text{n \choose k}$" giving us $\binom{n}{k}$.

The enhanced readability of the notation $\binom{n}{k}$ becomes rather obvious in more complex expressions. Compare for instance formula (5.32) in Concrete Mathematics by R. L. Graham, D. E. Knuth and O. Patashnik which is stated for integers $l,m,n$; $n\geq 0$ as \begin{align*} \sum_{j,k}(-1)^{j+k}\binom{j+k}{k+l}\binom{r}{j}\binom{n}{k}\binom{s+n-j-k}{m-j}=(-1)^l\binom{n+r}{n+l}\binom{s-r}{m-n-l} \end{align*} with the representation \begin{align*} \sum_{j,k}(-1)^{j+k}\,_{j+k}C_{k+l}\,_rC_j\,_nC_k\,_{s+n-j-k}C_{m-j}=(-1)^l\,_{n+r}C_{n+l}\,_{s-r}C_{m-n-l} \end{align*}

Answer 2

There is no difference at all. Personally I prefer using the shortest notation aCb for obvious reasons.

Answer 3

In books printed before $\LaTeX$ came to be widely used, setting ${n \choose r}$ into type was difficult (and expensive if used throughout a book). I 'know' this because of conversations with editors of math books over the years. By contrast, typesetting is not difficult for the variants of $C_r^n,$ including $C(n, r)$ not yet mentioned in this discussion.

Nowadays, I think there is a trend to use ${n \choose r},$ except for authors who have a strong preference for C-notations and those writing in Microsoft Word and trying to avoid using its 'equation editor'.

Curiously, a generally-accepted convention for permutations $P_r^n = r!{n \choose r}$ does not seem to have emerged. Feller used $(n)_r$ in his famous probability book, which one would have thought might have set a trend 60 years ago, but apparently not. His book may have set a record for the density per page of big parentheses for various uses of ${n \choose r}$-notation--even before $\LaTeX.$