Question

A professor packs her collection of 40 issues of a mathematics journal in four boxes with 10 issues per box. How many ways can she distribute the journals if

a) each box is numbered, so that they are distinguishable?

b) the boxes are identical, so that they cannot be distinguished?

a) each box is numbered, so that they are distinguishable?

Short answer

Therefore, there are\(4,705,360,871,073,570,227,520\)ways to distribute the journals if the boxes are distinguishable.

Step-by-step solution

Step 1: Assumptions and substitution

The collection contains forty issues of a journal, while there are ten issues per box among four boxes.

\(\begin{array}{l}n = 40\\k = 4\end{array}\)

\(\begin{array}{l}{n_1} = 10\\{n_2} = 10\\{n_3} = 10\\{n_4} = 10\end{array}\)

Step 2: Further simplification

Distributing n distinguishable objects into k distinguishable boxes such that \({n_i}\) objects are placed in box i (\(i = 1,2,3,4,5\)) can be done in \(\frac{{n!}}{{{n_1}!{n_2}!......{n_k}!}}\) ways\(\begin{array}{l}\frac{{n!}}{{{n_1}!{n_2}!......{n_k}!}} = \frac{{40!}}{{10!10!10!10!}}\\ = 4,705,360,871,073,570,227,520\end{array}\)

Therefore, there are\(4,705,360,871,073,570,227,520\)ways to distribute the journals if the boxes are distinguishable.

b) the boxes are identical, so that they cannot be distinguished?

Answer

Therefore, there are\(196,056,702,961,398,759,480\)ways to distribute the journals if the boxes cannot be distinguished.

Step 1(b): Assumptions and substitution

Since there are \(4!\) ways to order the 4 boxes, each way to distribute the journals if the boxes are indistinguishable than corresponds with\(4!\) ways to distribute the journals if the boxes are distinguishable (as simply recording the boxes will lead to the same way if the boxes will lead to the same way if the boxes are indistinguishable).

The number of ways to distribute the journals if the boxes are indistinguishable is then the number of ways to distribute the journals if the boxes are distinguishable( calculated in part (a) divided by the number of ways to order the boxes)

\(\begin{array}{l}\frac{{result\;part(a)}}{{4!}} = \frac{{40!}}{{4!10!10!10!10!}}\\ = \frac{{4,705,360,871,073,570,227,520}}{{24}}\\ = 196,056,702,961,398,759,480\end{array}\)

Step 2(b): Further simplification

Distributing n distinguishable objects into k distinguishable boxes such that \({n_i}\) objects are placed in box i (\(i = 1,2,3,4,5\)) can be done in \(\frac{{n!}}{{{n_1}!{n_2}!......{n_k}!}}\)ways\(\begin{array}{l}\frac{{n!}}{{{n_1}!{n_2}!......{n_k}!}} = \frac{{40!}}{{10!10!10!10!}}\\ = 4,705,360,871,073,570,227,520\end{array}\)

Therefore, there are\(196,056,702,961,398,759,480\)ways to distribute the journals if the boxes are indistinguishable.