In how many ways can 8 people stand in a line | vedclass

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Question

(A) Let the positions be $1, 2, 3, 4, 5, 6, 7, 8$. If $A$ is at position $i$,then $B$ must be at position $i+3$ (to have two people between them). The

Vedclass · Accepted Answer

$60 	imes 5!$

Vedclass · Answer

(A) Let the positions be $1, 2, 3, 4, 5, 6, 7, 8$. If $A$ is at position $i$,then $B$ must be at position $i+3$ (to have two people between them). The possible pairs of positions $(i, i+3)$ are $(1, 4), (2, 5), (3, 6), (4, 7), (5, 8)$. There are $5$ such pairs. Since $A$ and $B$ can interchange their positions,there are $5 	imes 2 = 10$ ways to place $A$ and $B$. The remaining $6$ people can be arranged in the remaining $6$ positions in $6!$ ways. Total ways $= 10 	imes 6! = 10 	imes 6 	imes 5! = 60 	imes 5!$.

In how many ways can $8$ people stand in a line such that there are always exactly two people between two specific people $A$ and $B$?

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