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Enumerating number of solutions to an equation

Determine the number of solutions of the equation $x_1+x_2+\dots+x_{10} = 100$ in positive integers not exceeding $30$.

Hint: First, find how many solutions this equation has if all $x_i$ are positive integers and one of them, say $x_1$, is constrained to be $> 30$.

To do this, consider numbers $x_1 − 30, x_2, x_3, \dots, x_{10}$.

Second, for a subset $S$ of $\{1, 2, \dots , 10\}$ find how many solutions the equation has if all $x_i$ are positive integers and in addition $x_j > 30$ for all $j \in S$.

Third, apply the Inclusion-Exclusion Principle. Notice what happens if $|S| \ge 4$.

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    The method is fully explained in [this answer](http://math.stackexchange.com/a/203839/12042) to a similar problem, and a more general result may be found in [this answer](http://math.stackexchange.com/a/146489/12042).2012-10-29

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