Question

There are two works each of $3$ volume, and two works each of $2$ volumes; in how many ways can the $10$ books be placed on a shelf so that volumes of the same work are not separated?

Answer 1

Let us consider the four works as four groups. Group $I$ and $II$ have 3 works each and groups $III$ and $IV$ have 2 volumes each.

Number of ways of arranging $4$ groups in a shelf is ${}^4{P}_4=4!=24\text{ways.}$

Volumes in each groups can themselves be arranged.

Group $I$ and $II$ can be arranged each in ${}^3{P}_3=3!=6\text{ways.}$

Group $III$ and $IV$ can be arranged each in ${}^2{P}_2=2!=2\text{ways.}$

Hence, total number of arrangemnts $=24\times 6\times 6\times 2\times 2=3456\text{ways.}$