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The stars and bars method visualizes nonnegative integer solutions to the equation $x_1+\cdots+x_n=r$. It represents the total sum $r$ as $r$ identical stars, which are separated into $n$ groups by $n-1$ bars.

Conditions: The variables $x_1, \dots, x_n$ are nonnegative integers.; The sum of the variables equals $r$.; Empty groups are allowed, meaning a variable can be zero.

The stars and bars method visualizes nonnegative integer solutions to the equation $x_1+\cdots+x_n=r$. It represents the total sum $r$ as $r$ identical stars, which are separated into $n$ groups by $n-1$ bars.

Conditions: The variables $x_1, \dots, x_n$ are nonnegative integers.; The sum of the variables equals $r$.; Empty groups are allowed, meaning a variable can be zero.