® The number of combinations of n different things taken r at a time, denoted by n C r , is given by n C r = ! ! ! r ( n r ) , ≤ r ≤ n .
Historical Note The concepts of permutations and combinations can be traced back to the advent of Jainism in India and perhaps even earlier. The credit, however, goes to the Jains who treated its subject matter as a self-contained topic in mathematics, under the name Vikalpa . Among the Jains, Mahavira , (around ) is perhaps the world’s first mathematician credited with providing the general formulae for permutations and combinations. In the 6th century B.C., Sushruta, in his medicinal work, Sushruta Samhita , asserts that combinations can be made out of different tastes, taken one at a time, two at a time, etc.
Pingala , a Sanskrit scholar around third century B.C., gives the method of determining the number of combinations of a given number of letters, taken one at a time, two at a time, etc. in his work Chhanda Sutra . Bhaskaracharya (born ) treated the subject matter of permutations and combinations under the name Anka Pasha in his famous work Lilavati. In addition to the general formulae for n C r and n P r already provided by Mahavira, Bhaskaracharya gives several important theorems and results concerning the subject.
Outside India, the subject matter of permutations and combinations had its humble beginnings in China in the famous book I–King (Book of changes). It is difficult to give the approximate time of this work, since in B.C., the emperor had ordered all books and manuscripts in the country to be burnt which fortunately was not completely carried out. Greeks and later Latin writers also did some scattered work on the theory of permutations and combinations. Some Arabic and Hebrew writers used the concepts of permutations and combinations in studying astronomy.
Rabbi ben Ezra, for instance, determined the number of combinations of known planets taken two at a time, three at a time and so on. This was around . It appears that Rabbi ben Ezra did not know PERMUTATIONS AND COMBINATIONS the formula for n C r . However, he was aware that n C r = n C n – r for specific values n and r .
In , Levi Ben Gerson , another Hebrew writer came up with the formulae for n P r , n P n and the general formula for n C r . The first book which gives a complete treatment of the subject matter of permutations and combinations is Ars Conjectandi written by a Swiss, Jacob Bernoulli ( – ), posthumously published in . This book contains essentially the theory of permutations and combinations as is known today.