Combinations, Choose .... Notes on History and Etymology of Math Terms

  Choose

The creation of the expression "N choose R" for a term of a binomial coefficient is credited to Richard Guy sometime around 1950. Although the symbol for the number of ways of selecting a group of r distinct items from a group of n distinct items still varies, the most consistent usage today is . The symbol is read as "N choose r". The number and symbol is also called the Combination symbol, and is sometimes read as "The combinations of n things taken r at a time." Other symbols still used include and the less common . It is also very common to use a Capital C between the values of n and r with the values subscripted nCr, and many calculators still use a notation such as nCr with the numbers on the same line level with the C. Mathematica uses "Binomial[n,r]", and at one time I know that the TEX formatting language used (n\choose r}.

Most students first see the binomial coefficients as elements in the array (mis)named Pascal's Triangle

More information about computing binomial coefficients can be found in the page on combinations and permutations. The use of the term Binomial Coefficients comes from the fact that the numbers are the same as the coefficients of each term when a binomial, such as (x+y) is raised to a power. For example the expansion of (x+y)4 gives the five terms in the fifth row of Pascal's triangle.

In Early July of 2004 I received a note from Matthew Hubbard, the curator of Pascal's Triangle From Top to Bottom. In it he informed me that I had failed to include, in particular, the contribution of India to the study of the arithmetic triangle. A quick visit to his web site led me to :

The idea of taking "six tastes one at a time, two at a time, three at a time, etc." was written down correctly in India 300 years before the birth of Christ in a book called the Bhagabati Sutra, a text from the Jainist religion; this gives the subcontinent of India the distinction of being the earliest civilization to have an understanding of the binomial coefficients in their combinatorial form "n choose k" in a text that survives to this day.

The site contains much additional material about Indian study of the triangle, and other information that makes it well worth a visit.

Matthew also called me to task for my suggestion above that the use of "Pascal's Triangle" was somehow inappropriate. He wrote in justification of the term, " One of the reasons I wrote is the idea of misnomy in mathematics; you put the word (mis)named in front of Pascal's Triangle. While it is certainly true that many, many people had studied the binomial coefficients prior to Pascal, his work is honored because it was read by people who came after him, most notably Monmort and deMoivre, who credited Pascal's Treatise in their works several decades later. Moreover, it is worth reading, as Pascal finds many identities in the triangle that no one before him had written down.
It's too late to get the world to call it Pingala's Triangle, and I fully appreciate the desire of civilizations to honor their own, but I think if anybody's name is going to be linked to this famous array, Blaise Pascal is as good a candidate as any and significantly better than most."

My thanks to Matt for the additional material on the Indian contribution, and for helping to insure a balance of credit where credit is due, and certainly Pascal is due much credit for his exposure of many aspects of the triangle, by whatever name it is called.