Find the number of solutions of the equation:
![\[x_1+x_2+\cdots + x_{100} = 2017\]](https://www.math.tolaso.com.gr/wp-content/ql-cache/quicklatex.com-244b3f8953ac7cd0fde39575653dd6d6_l3.png "Rendered by QuickLaTeX.com")
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
![\[\bullet \bullet \bullet \bullet \bullet \mid \bullet \bullet \mid \mid \bullet\]](https://www.math.tolaso.com.gr/wp-content/ql-cache/quicklatex.com-463a7902998ddff5f9c0787f8af6afe0_l3.png "Rendered by QuickLaTeX.com")
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
![\[\binom{1917+99}{99} = \binom{2016}{99}\]](https://www.math.tolaso.com.gr/wp-content/ql-cache/quicklatex.com-98baaaf857de801f5b304ae7670e22e5_l3.png "Rendered by QuickLaTeX.com")
solutions.
Comment: In general the equation
![\[x_1+x_2+\cdots+x_m=n\]](https://www.math.tolaso.com.gr/wp-content/ql-cache/quicklatex.com-a033744aaab0d283f9020f47056f956c_l3.png "Rendered by QuickLaTeX.com")
has 
solutions in the positive integers and 
solutions in the non negative integers.