Question

In how many ways can 19 identical books on English and 17 identical books on Hindi be placed in a row on a shelf so that two books on Hindi may not be together ?

Answer 1

In order that two books on Hindi are never together, we must place these books as under :

XEXEXEX...XEX,

where E denotes the position of an English book and X that of a Hindi book.

Since there are 19 books on English, the number of cross marks are therefore, 20.

Since all books on English are identical, they can be placed in only 1 way at the places marked E.

Number of ways of placing 17 identical books on Hindi at 20 places, each marked $X{=}^{20}{C}_{17}={}^{20}{C}_3=\frac{20\times 19\times 18}{3\times 2\times 1}=1140$

Hence, the required number of ways = $(1\times 1140)=1140$