Q. 234.3( 4 Votes )

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 :

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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