Some simple discrete probabilistic processes and a problem of statistical inference are indicated where occur, in a natural way, numberss (n, k) which are the absolute value of the Stirling numbers of the first kind. We propose that the following notation for the Stirling numbers of the first kind:\(\left( {x + n - 1} \right)_n = \sum\limits_{k = 0}^n { s\left( {n, k} \right) x^k } \), should be standardized. This notation has many advantages, for example for the derivation of properties of the Stirling numbers of the first kind with combinatoric methods. Here in a purely combinatorial way a recurrence relation for thes(n, k) is obtained which is not mentioned, as a rule, among their properties and is presumably new. Starting from this we obtain a new explict expression for thes(n, k), «a closed form» according to Abramowitz and Stegun, whithout the Stirling numbers of the second kind, which is of course as well solution of the «classical» equation of partial differences for thes(n, k).