Five balls of different colours are to be pla | vedclass

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(C) Let the boxes be marked as $A, B, C$. We have to ensure that no box remains empty and all five balls are placed.There are two possible distributio

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$150$

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(C) Let the boxes be marked as $A, B, C$. We have to ensure that no box remains empty and all five balls are placed.There are two possible distributions of $5$ balls into $3$ boxes such that no box is empty:$(i)$ Two boxes contain $1$ ball each and the $3^{rd}$ box contains $3$ balls.The number of ways to choose the balls is $^5C_1 	imes ^4C_1 	imes ^3C_3 = 5 	imes 4 	imes 1 = 20$.Since the box containing $3$ balls can be any of the $3$ boxes,the total ways for this case is $20 	imes 3 = 60$.$(ii)$ Two boxes contain $2$ balls each and the $3^{rd}$ box contains $1$ ball.The number of ways to choose the balls is $^5C_2 	imes ^3C_2 	imes ^1C_1 = 10 	imes 3 	imes 1 = 30$.Since the box containing $1$ ball can be any of the $3$ boxes,the total ways for this case is $30 	imes 3 = 90$.Therefore,the total number of ways is $60 + 90 = 150$.

Five balls of different colours are to be placed in three boxes of different sizes. Each box can hold all five balls. In how many ways can we place the balls so that no box remains empty?

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