• question_answer 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 [IIT 1981] A) 50 B) 100 C) 150 D) 200

Let the boxes be marked as $A,\ B,\ C$. We have to ensure that no box remains empty and in all five balls have to put in. There will be two possibilities. (i) Any two containing one and ${{3}^{rd}}$ containing 3. $A$(1) $B$(1) $C$ (3) $^{5}{{C}_{1}}{{.}^{4}}{{C}_{1}}{{.}^{3}}{{C}_{3}}=5\ .\ 4\ .\ 1=20$. Since the box containing 3 balls could be any of the three boxes $A,\ B,\ C$. Hence the required number is = $20\times 3=60$. (ii) Any two containing 2 each and ${{3}^{rd}}$containing 1. $A$(2) $B$(2) $C$ (1) $^{5}{{C}_{2}}{{.}^{3}}{{C}_{2}}{{.}^{1}}{{C}_{1}}=10\times 3\times 1=30$ Since the box containing 1 ball could be any of the three boxes $A,\ B,\ C$. Hence the required number is = $30\times 3=90$. Hence total number of ways are = $60+90=150$.