 The second part expands on enumerative combinatorics , or the systematic numeration of objects . It was in this part that two of the most important of the twelvefold ways — the permutations and combinations that would form the basis of the subject — were fleshed out , though they had been introduced earlier for the purposes of probability theory . He gives the first non @-@ inductive proof of the binomial expansion for integer exponent using combinatorial arguments . On a note more distantly related to combinatorics , the second section also discusses the general formula for sums of integer powers ; the free coefficients of this formula are therefore called the Bernoulli numbers , which influenced Abraham de Moivre 's work later , and which have proven to have numerous applications in number theory . 
