Question

Zoe has 4 blue, 5 red, and 3 green books to arrange on a shelf. In how many ways can the books be arranged if books of the same color must be grouped together?

Answer 1

If the books of same color must be grouped together, then there are 3!= 6 ways. let us see how

first we arrange all Blue books, then all Red, then all Green. let us call this arrangement BRG.

Also, we can arrange first all blue, then all green and then all red. This is BGR.

Similarly, we can also arrange in GBR, GRB, RGB, RBG manner. so, total 6 six ways.

Actually we have 3 empty places. For first we have choices from 3 colour books, for second we have remaining 2 colours to choose from and for the third place the remaining last colour. Hence, 321=6 ways. As nothing is given about wether the books of same colour are distinguishable or not, then this means books of same colour are similar. Hence, it does not matter in which way we arrange the books of same colour.