⭐ Stars and Bars

Partitioning problem

Determine the number of ways to partition indistinguishable objects into distinguishable bins.

Theory:

The number of ways to distribute $n$ indistinguishable objects into $k$ distinguishable bins is given by:

$$C(n+k-1,k-1)=\frac{(n+k-1)!}{(k-1)!n!}$$

Where:

• $n$ : Number of indistinguishable objects
• $k$ : Number of distinguishable bins

If each bin must contain at least one object, the formula becomes:

$$C(n-1,k-1)$$