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(A) First,arrange the $6$ distinct white roses in a circle. The number of ways to arrange $n$ distinct items in a circle is $(n-1)!$. So,$6$ white ros

Vedclass · Accepted Answer

$43200$

Vedclass · Answer

(A) First,arrange the $6$ distinct white roses in a circle. The number of ways to arrange $n$ distinct items in a circle is $(n-1)!$. So,$6$ white roses can be arranged in $(6-1)! = 5! = 120$ ways.There are $6$ gaps created between these $6$ white roses. We need to place $6$ distinct red roses in these $6$ gaps so that no two red roses are together. The number of ways to arrange $6$ distinct red roses in $6$ gaps is $6! = 720$ ways.Since it is a garland,the clockwise and anti-clockwise arrangements are considered the same. Therefore,we divide the total number of linear arrangements by $2$.The total number of ways $= \frac{5! 	imes 6!}{2} = \frac{120 	imes 720}{2} = \frac{86400}{2} = 43200$.

The number of different ways of preparing a garland by using $6$ distinct white roses and $6$ distinct red roses such that no two red roses come together is

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