Combinations with Repetition

${}_{n+r-1}{C}_r$ or ${}_{n+r-1}{C}_{n-1}$ or $\frac{(r+n-1)!}{r!(n-1)!}$

Intuition

Best visualized if you think of the item pool being selected like: Where:

• Elements are arranged left to right in the boxes starting from element id 1 to element id n
• Bars tell us when to stop placing our current element and move onto the next element Note that there can be at most $r-1$ bars as you cannot have a partition with no category. Then, if we add these bars into our element pool, you get: Then the combinations are: ${}_{n+r-1}{C}_r$

Stars and Bars for Sums Of Numbers

${}_{\text{sum+\#divisors}-1}{C}_{\text{\#divisors}-1}$ Used to find the number of solutions to an equation.

Example

If for example you have $x_1+x_2+x_3+x_4=n$, to find all possible solutions you can do: ${}_{n_1+4-1}{C}_3$