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### Group combinations (Division into groups)

A set of objects can be divided into two or more groups containing different or equal number of objects. For example, if a group of 20 different balls is divided into two groups, one group containing 7 and the other containing 13 balls, then in how many ways is this division possible?

The above question is solved by the following formula:
If p + q objects are to be divided into two groups containing p and q objects respectively, then this can be done in the following different ways:
\frac{(p + q)!}{p! \cdot q!}
For example, if 20 (~ p + q) objects are to be divided into two groups of 13 ( ~ p) and 7 ( ~ q) objects respectively, then the total number of ways of this division are
\frac { 20! } { 13! \cdot 7! }
Simplifying this, you obtain 77520. Thus there are 77520 different ways of dividing a group of 20 different objects into two groups containing 13 and 7 objects respectively.

The above formula can be applied when dividing a group of objects into any number of groups or divisions. For example, if dividing p + q + r objects into three groups consisting of p, q and r objects, then the total number of ways of doing so are
\frac{(p + q + r)!}{p! \cdot q! \cdot r!}