Question

In how many ways 11 identical toys be placed in 3 distinct boxes such that no box is empty?

Answer 1

In 45 ways 11 identical toys be placed in 3 distinct boxes such that no box is empty.

• As no box can be empty so all the boxes have at least one toy in them. Thus there is choice only with respect to the remaining 11-3 = 8 toys.

• Let the boxes be named X,Y and Z.  If X has 8 toys, then Y can be filled in only one way.  If X has 7 toys, then Y can be filled in 2 distinct ways [0 or 1 toys].  If X has 6 toys, then Y can be filled in 3 distinct ways [0,1 or 2 toys] and so on.

• We are not interested about the box Z as the number of toys in the third box is fixed as if we select how many toys are the first two boxes.

• Thus, the total number of ways we can fill 8 toys in three boxes is 1+2+3…+8+9=(9*10)/2 = 45 ways.