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 third box contains $3$ balls.Number of ways to choose balls for the boxes: $\binom{5}{1} 	imes \binom{4}{1} 	imes \binom{3}{3} = 5 	imes 4 	imes 1 = 20$.Since the box containing $3$ balls can be any of the $3$ boxes ($A, B,$ or $C$),the total ways for this case = $20 	imes 3 = 60$.$(ii)$ Two boxes contain $2$ balls each,and the third box contains $1$ ball.Number of ways to choose balls for the boxes: $\binom{5}{2} 	imes \binom{3}{2} 	imes \binom{1}{1} = 10 	imes 3 	imes 1 = 30$.Since the box containing $1$ ball can be any of the $3$ boxes ($A, B,$ or $C$),the total ways for this case = $30 	imes 3 = 90$.Total number of ways = $60 + 90 = 150$.

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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