COMBINATIONS with REPETITION - DISCRETE MATHEMATICS

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.

To account for a variable's lower bound, you substitute a new variable to shift the bound to zero. For example, if $x_3 \ge 2$, you define $x_3' = x_3 - 2$.

Conditions: One or more variables have a lower bound greater than zero.; The other variables are nonnegative.; The equation is of the form $x_1+\cdots+x_n=r$.

To calculate the number of combinations with repetition, where order is ignored and repetition is allowed, you use the formula $\binom{n+r-1}{r}$. This counts the number of ways to select $r$ objects from $n$ distinct types.

Conditions: Repetition is allowed.; Order does not matter.; There are $n$ distinct types of objects.; You are selecting exactly $r$ objects.