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

Given : There are two works each of 3 volumes and two works each of 2 volumes.

To find : Number of ways in which these books can be arranged in a shelf provided volumes of the same work are not separated.

Let w₁, w₂, w₃, w₄, are four works 

w₁ has n₁, n₂, n₃ as volumes 

w₂ has m₁, m₂, m₃ as volumes 

w₃ has a₁, a₂ as volumes 

w₄ has b₁, b₂ as volumes 

Now,

Firstly we have to arrange these 4 works like w₂ w₃ w₁ w₄ or w₁ w₂ w₄ w₃ 

This can be done in 4! ways 

Now,

We have to separately arrange volumes of these 4 works w₁ has 3 volumes which can be arranged like n₂ n₁ n₃ or n₃ n₁ n₂ 

Volumes of w₁ can be arranged in 3! ways 

Similarly, 

Volumes of w₂ can be arranged in 3! ways Volumes of w₃ can be arranged in 2! ways 

Volumes of w₄ can be arranged in 2! Ways 

∴ Total number of ways = 4! × 3! × 3! × 2! × 2! 

= 24 × 6 × 6 × 2 × 2 

= 3456 

Hence,

The total number of ways in which these 10 books be placed on a shelf so that the volumes of the same work are not separated are 3456