# 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?

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

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