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

Question

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

Tardigrade Admin · Accepted Answer

Since, each box can hold five balls. ∴ Number of ways in which balls could be distributed so that none is empty, are (2, 2, 1) or (3, 1, 1). i.e.,, (5C2 3C2 1C1 + 5C3 2C1 1C1) × 3! = (30 + 20) × 6 = 300

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

Since, each box can hold five balls.
$∴$ Number of ways in which balls could be distributed so that none is empty, are $(2, 2, 1)$ or $(3, 1, 1)$ .
i.e.,, $(^{5} C_{2}^{3} C_{2}^{1} C_{1} +^{5} C_{3}^{2} C_{1}^{1} C_{1}) \times 3!$
$= (30 + 20) \times 6 = 300$

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

Since, each box can hold five balls. ∴ Number of ways in which balls could be distributed so that none is empty, are (2,2,1) or (3,1,1). i.e.,, (5C2​3C2​1C1​+5C3​2C1​1C1​)×3! =(30+20)×6=300

Since, each box can hold five balls.
$∴$ Number of ways in which balls could be distributed so that none is empty, are $(2, 2, 1)$ or $(3, 1, 1)$ .
i.e.,, $(^{5} C_{2}^{3} C_{2}^{1} C_{1} +^{5} C_{3}^{2} C_{1}^{1} C_{1}) \times 3!$
$= (30 + 20) \times 6 = 300$