Six persons A, B, C, D, E and F are to be sea | vedclass

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Question

(A) Fix $A$ at one position. Let the positions be $1, 2, 3, 4, 5, 6$ in clockwise order,with $A$ at position $1$. The immediate right of $A$ is positi

Vedclass · Accepted Answer

$18$

Vedclass · Answer

(A) Fix $A$ at one position. Let the positions be $1, 2, 3, 4, 5, 6$ in clockwise order,with $A$ at position $1$. The immediate right of $A$ is position $6$ (since they face the centre).Case $1$: $E$ is at position $6$. Then $E$ must have $F$ or $D$ at position $5$.Subcase $1.1$: $F$ is at position $5$. The remaining $3$ persons $(B, C, D)$ can be arranged in the remaining $3$ positions in $3! = 6$ ways.Subcase $1.2$: $D$ is at position $5$. The remaining $3$ persons $(B, C, F)$ can be arranged in the remaining $3$ positions in $3! = 6$ ways.Total for Case $1 = 6 + 6 = 12$ ways.Case $2$: $F$ is at position $6$. Then $E$ must have $F$ or $D$ on his immediate right. Since $F$ is at position $6$,$E$ cannot be at position $5$ (because $F$ is at $6$,not $5$). Thus,$E$ must be at some other position $k$ such that position $k-1$ (clockwise) is $F$ or $D$.By checking the remaining positions,we find $6$ valid arrangements for Case $2$.Total ways $= 12 + 6 = 18$.

Six persons $A, B, C, D, E$ and $F$ are to be seated at a circular table facing towards the centre. Find the number of ways this can be done if $A$ must have either $E$ or $F$ on his immediate right and $E$ must have either $F$ or $D$ on his immediate right.

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