Question

The number of ways in which 6 Telugu, 4 Hindi and 3 English books be placed in a row on a shelf so that the books of the same subject should remain together is
A) $6!4!3!$
B) $3!6!4!3!$
C) $\dfrac{{13!}}{{6!4!3!}}$
D) $\dfrac{{13!}}{{6!4!3!3!}}$

Answer 1

Hint:
The number of ways in which $n$ items can be placed irrespective of any condition is given by $n!$. Consider the books of Telugu, Hindi and English as three different items and find the ways of arranging them. Multiply the result by the individual arrangements within the item.

Complete step by step solution:
We are given that the total of 6 Telugu, 4 Hindi and 3 English books are present to be arranged in a row on a shelf.
Thus a total of $6 + 4 + 3$, that is 13 books have to be arranged on the row on a shelf.
The number of ways in which $n$ items can be placed irrespective of any condition is given by $n!$.
If no condition was to be imposed on the arrangement of the books on the row, then the total possible arrangements would be equal to $13!$.
However, it is given that the books of the same subject must be placed together.
Since each book of the same subject will be placed next to each other, each subject’s book can be together treated as an item.
Thus the problem is therefore reduced to arranging the three different subject books on the row.
The total ways the three subject’s books can be placed is therefore $3!$.
However, books of the same subject can also be placed in any order.
The number of ways that the Telugu books can be arranged is $6!$.
 The number of ways that the Hindi books can be arranged is $4!$.
The number of ways that the English books can be arranged is $3!$.
Thus the total number of ways that the 6 Telugu, 4 Hindi and 3 English books be placed in a row on a shelf so that the books of the same subject should remain together is $3! \times 6! \times 4! \times 3!$.

Thus the option B is the correct answer.

Note:
The number of ways in which $n$ items can be placed irrespective of any condition is given by $n!$. The total number of arrangements of $n$ items is equal to the $n!$ multiplied the number of arrangements of each item possible.