In how many ways can the letters of the word | vedclass

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Question

(A) The word $BALLOON$ contains $7$ letters in total,where $L$ appears $2$ times and $O$ appears $2$ times.The total number of arrangements of these $

Vedclass · Accepted Answer

$900$

Vedclass · Answer

(A) The word $BALLOON$ contains $7$ letters in total,where $L$ appears $2$ times and $O$ appears $2$ times.The total number of arrangements of these $7$ letters is given by $\frac{7!}{2! 	imes 2!} = \frac{5040}{4} = 1260$.To find the number of arrangements where the two $Ls$ come together,we treat the two $Ls$ as a single unit $(LL)$. Now,we have $6$ units to arrange: $(LL), B, A, O, O, N$. In these $6$ units,$O$ repeats $2$ times.The number of arrangements where the two $Ls$ are together is $\frac{6!}{2!} = \frac{720}{2} = 360$.Therefore,the number of ways in which the two $Ls$ do not come together is the total arrangements minus the arrangements where they are together:$1260 - 360 = 900$.

In how many ways can the letters of the word $BALLOON$ be arranged so that two $Ls$ do not come together?

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