Question

In how many different ways can all of 5 identical balls be placed in the cells shown below such that each row contains at least 1 ball?

(a) 64 (b) 81 (c) 84 (d) 108

Answer 1

(d) There can be two cases : 

Case I : When 1 row contains 3 balls and rest two contains 1 ball each. 

Now, the row which contains 3 balls can be selected out of the 3 rows in ³C₁ ways and in this row number of ways of arrangement = ³C₃. In other two rows, number of ways of arrangement in each = ³C₁. 

Thus, number of ways for case I = ³C₁ × ³C₃ × ³C₁ × ³C₁ 

= 3 × 1 × 3 × 3 = 27

 Case II : When 1 row contains 1 ball and rest two rows contain 2 balls each. 

This row, containing 1 ball can be selected in ³C₁ ways and number of ways of arrangement in this row = ³C_1. In other two rows, containing 2 balls each, number of ways of arrangement in each = ³C₂. 

Thus, number of ways for case II = ³C₁ × ³C₁ × ³C₂ × ³C₂ = 3 × 3 × 3 × 3 = 81 

As, either of these two cases are possible, hence total number of ways = case I or case II = 27 + 81 = 108.