Please submit your concern

Question:

In how many ways can 7 identical erasers be distributed among 4 kids in such a way that each kid gets at least one eraser but nobody gets more than 3 erasers?

Explanation:

After giving one eraser to each of the 4 kids, there are 3 left.

They can split 2, 1 or 1, 1, 1. (No kid can get 4)

Number of ways of 2, 1 split = ⁴P₂
Number of ways of 1, 1, 1 split = ⁴C₃

There are ⁴P₂ + ⁴C₃, i.e., 16 ways of distributing the erasers.

Alternately,
Let the number of erasers given to the 4 kids be w, x, y, z.

w + x + y + z = 7.

After giving at least 1 eraser to each, 3 erasers will be left.

w' + x' + y' + z' = 3

The number of positive integral solutions is ^{3 + 4 - 1}C_{4 - 1}, i.e. 20. This includes (4,1,1,1); (1,4,1,1); (1,1,4,1); (1,1,1,4).

The required number = 20 – 4 = 16.

Hence, option (a).