Question

In how many ways can 21 English and 19 Hindi books be placed in a row so that no two Hindi are together?
(a) 1540
(b) 1450
(c) 1504
(d) 1405

Answer 1

Hint: To solve this question, we will first arrange all the English books. Then we will determine how many places are created from Hindi books to be placed. After finding out the number of places available, we will select 19 Hindi books and place them in these places so that no Hindi book is together.

Complete step-by-step answer:
The question demands that we have to arrange Hindi books so that no two of them are together. For this, it is necessary that between any two Hindi books, there must be at least one English book and no two Hindi books should be placed just after another. For arranging the books in this way, we will first arrange English books. There are 21 English books, so the number of places in which Hindi books can be placed is 22 because they are arranged as shown.
\[\underline{H}E\underline{H}E\underline{H}E\underline{H}E\underline{H}......\]
Here, H represents Hindi books and E represents English books. Now, when an English book is placed, 2 places are available. When 2 English books are placed, 3 places are available. Similarly, when 21 English books are placed, 22 places are available. Now, we will arrange Hindi books in these places. When Hindi books are arranged in these places, no two Hindi books would be together. Thus the number of ways by which we can place 19 Hindi books in these 22 places is \[^{22}{{C}_{19}}.\]
Now, here we are going to use the formula as shown:
\[^{n}{{C}_{r}}={{\text{ }}^{n}}{{C}_{n-r}}\]
After applying this formula, we will get:
Number of arrangements \[={{\text{ }}^{22}}{{C}_{\left( 22-19 \right)}}={{\text{ }}^{22}}{{C}_{3}}\]
\[=\dfrac{22!}{19!3!}\]
= 1540
Hence, option (a) is correct.

Note: We can solve this question because here English books are greater in number than Hindi books. If the number of Hindi books is greater than the number of English books, then it is not possible to arrange them, so that no two of them are together. For example, when English books 5 and the number of Hindi books are 10, then the number of places will be 6 and we will have to arrange more than one Hindi book at that place and that will violate the condition given in the question.