Question

In how many ways can $21$ English and $19$ Hindi books be placed in a row so that no two Hindi books are together?

• A. $1540$
• B. $1450$
• C. $1504$
• D. $1405$

Answer 1

The correct option is A

$1540$

Explanation for the correct option:

Finding the ways in which no two Hindi books are together.

Since number of book of English = $21$

Number of Hindi book = $19$

According to question no book comes together

To place $21$ English books in we have $22$ gaps.

Therefore, $19$ Hindi books can be kept in a row when no two Hindi books are together.

Using the formulae of combination

$\begin{array}{rcl}{}^{22}c_{19}& =& \frac{22!}{19!(22-19)!}\\ & =& \frac{22\times 21\times 20\times 19!}{19!\times 3!}\\ & =& \frac{22\times 21\times 20}{3\times 2\times 1}\\ & =& 1540\end{array}$${}^{\mathbf{n}}\mathit{c}_{\mathbf{r}\mathbf{}}\mathbf{=}\mathbf{}\frac{\mathbf{n}\mathbf{!}}{\mathbf{r}\mathbf{!}\mathbf{(}\mathbf{n}\mathbf{-}\mathbf{r}\mathbf{)}\mathbf{!}}$

Hence, the correct option is $\left(A\right)$