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# Number of solutions

Find the number of solutions of the equation:

in the positive integers.

Solution

We set . It suffices to find the number of solutions of the equation

(1)

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

solutions.

Comment: In general the equation

has solutions in the positive integers and solutions in the non negative integers.