Find the number of solutions of the equation:
in the positive integers.
We set . It suffices to find the number of solutions of the equation
in the non negative numbers. We represent each sum of non negative integers with a sequence of dots () followed by a vertical bar (), after dots another one vertical bar etc, till we place the last dots ( without the vertical bar at the end.) For example the sum can be represented as
We note that every solution of matches a sequence that has dots in total and vertical bars. Conversely, every such sequence matches a solution of .
Thus, in total there are
Comment: In general the equation
has solutions in the positive integers and solutions in the non negative integers.