Kenneth Rosen Edition 7 Exercise 6.5 Question 56 (Page No. 434)

Question

How many ways are there to pack eight identical DVDs into five indistinguishable boxes so that each box contains at least one DVD?

Answer 1

this is the IOIB (identical Objects, Identical Boxes) problem. According to the problem, each box has at least one DVD.

Let each box be labelled, B1, B2, B3, B4. B5.

$\therefore$  We need to find the number of partitions of 8 into 5 parts where $B_{i}$ $\geqslant $ 1

There are 3 ways

8 = 1+1+1+1+4 

8 = 1+1+2+2+2

8 = 1+1+1+2+3

(NOTE: since all the objects and boxes are same, all other permutations of the partitions will be same)

Thus, there are 3 ways to pack eight identical DVDs into five indistinguishable boxes so that each box contains at least one DVD.