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(B) There are $5$ positions to be filled by $5$ students.Position $2$ is fixed for the boy $SUNIL$.This leaves $4$ positions to be filled by the remai

Vedclass · Accepted Answer

$8$

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(B) There are $5$ positions to be filled by $5$ students.Position $2$ is fixed for the boy $SUNIL$.This leaves $4$ positions to be filled by the remaining $4$ students.Since $GITA$ and $NITA$ must always be adjacent,we treat them as a single unit $(GITA, NITA)$.Now,we have the unit $(GITA, NITA)$ and the $2$ other remaining students,making a total of $3$ entities to be arranged in the remaining $4$ positions.However,since $SUNIL$ is fixed at position $2$,we are effectively arranging these $3$ entities in the remaining $3$ available slots (positions $1, 3, 4, 5$ are available,but the unit occupies $2$ adjacent spots).Let the positions be $P_1, P_2, P_3, P_4, P_5$. $P_2$ is $SUNIL$.The remaining positions are $P_1, P_3, P_4, P_5$.The unit $(GITA, NITA)$ can occupy positions $(P_3, P_4)$ or $(P_4, P_5)$.Case $1$: Unit is at $(P_3, P_4)$. The other $2$ students can be arranged in $P_1$ and $P_5$ in $2!$ ways. The unit itself can be arranged as $(GITA, NITA)$ or $(NITA, GITA)$ in $2!$ ways. Total $= 2! 	imes 2! = 4$.Case $2$: Unit is at $(P_4, P_5)$. The other $2$ students can be arranged in $P_1$ and $P_3$ in $2!$ ways. The unit itself can be arranged in $2!$ ways. Total $= 2! 	imes 2! = 4$.Case $3$: Unit is at $(P_1, P_2)$ is impossible because $P_2$ is occupied by $SUNIL$.Wait,let's re-evaluate: The $4$ remaining students are $S_1, S_2, GITA, NITA$. $SUNIL$ is at $P_2$. The available spots are $P_1, P_3, P_4, P_5$.If we treat $(GITA, NITA)$ as one block $B$,we have $S_1, S_2, B$. These $3$ items can be arranged in the remaining spots such that $B$ is adjacent. The possible positions for $B$ are $(P_3, P_4)$ or $(P_4, P_5)$.For each position of $B$,the other $2$ students can be arranged in the remaining $2$ spots in $2!$ ways. The block $B$ can be arranged in $2!$ ways.Total arrangements $= 2 	imes (2! 	imes 2!) = 2 	imes 4 = 8$.

The principal wants to arrange $5$ students on the platform such that the boy $SUNIL$ occupies the second position and such that the girl $GITA$ is always adjacent to the girl $NITA$. How many such arrangements are possible?

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