In how many ways can 21 English and 19 Hindi | vedclass

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Question

(A) To ensure that no two Hindi books are together,we first arrange the $21$ English books in a row.These $21$ English books create $22$ possible gaps

Vedclass · Accepted Answer

$1540$

Vedclass · Answer

(A) To ensure that no two Hindi books are together,we first arrange the $21$ English books in a row.These $21$ English books create $22$ possible gaps (including the ends) where the Hindi books can be placed: $\bullet E_1 \bullet E_2 \bullet E_3 \bullet ... \bullet E_{21} \bullet$.Since there are $19$ Hindi books,we need to choose $19$ gaps out of the $22$ available gaps.The number of ways to choose these gaps is given by the combination formula $^{n}C_{r} = \frac{n!}{r!(n-r)!}$.Here,$n = 22$ and $r = 19$.Total ways = $^{22}C_{19} = ^{22}C_{22-19} = ^{22}C_{3}$.Calculating the value: $^{22}C_{3} = \frac{22 	imes 21 	imes 20}{3 	imes 2 	imes 1} = 22 	imes 7 	imes 10 = 1540$.Thus,there are $1540$ ways to arrange the books.

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

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