Question

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

Answer 1

Let $w_1,w_2,w_3,w_4$ are four works.

$w_1$ has $n_1,n_2,n_3$ as volumes.

$w_2$ has $m_1,m_2,m_3$ as volumes.

$w_3$ has $a_1,a_2$ as volumes.

$w_4$ has $b_1,b_2$ as volumes.

Now, firstly we have to arrange these 4 works.

This can be done in 4! ways.

Now, we have to separately arrange volumes of these 4 works.

Volumes of $w_1$ can be arranged in 3! ways.

Volumes of $w_2$ can be arranged in 3! ways.

Volumes of $w_3$ can be arranged in 2! ways.

Volumes of $w_4$ can be arrnaged in 2! ways.

Therefore,

Total number of ways = $4!\times 3!\times 3!\times 2!\times 2!=3456$