Question

Show that there aredifferent unordered selections of n objects of r different types that include at leastobjects of type one, q2 objects of type two,...,andobjects of type r

Short answer

There are \(C(n{\rm{ }} + {\rm{ }}r{\rm{ }} - {\rm{ }}{q_1}{\rm{ }} - {\rm{ }}{q_2}{\rm{ }}... - \;{q_r}\; - 1,{\rm{ }}n{\rm{ }} - {\rm{ }}{q_1}{\rm{ }} - {\rm{ }}{q_2}{\rm{ }} - ... - \;{q_r}\;)\) selections

Step-by-step solution

Step 1: Types of n and r objects

There are n objects of r different types.

There are at leastobjects of type one,

There are at leastof type 2, at leastof type r.

We select objects of type one, of type 2, of type r.

Step 2: Solution of the selection

The order of selection does not matter.

Let’s, consider the n and r objects

so, there arethat has to be selected from r different types.

Using the definition of combination with repetition allowed, we obtain that there are

\(C(n{\rm{ }} + {\rm{ }}r{\rm{ }} - {\rm{ }}{q_1}{\rm{ }} - {\rm{ }}{q_2}{\rm{ }}... - \;{q_r}\; - 1,{\rm{ }}n{\rm{ }} - {\rm{ }}{q_1}{\rm{ }} - {\rm{ }}{q_2}{\rm{ }} - ... - \;{q_r}\;)\) selections