Problem solving - 14

Larry, Michael, and Doug have five donuts to share. If any one of the men can be given any whole number of donuts from 0 to 5, in how many different ways can the donuts be distributed?

(A) 21
(B) 42
(C) 120
(D) 504
(E) 5040

Answer - A -- generally answer for such questions is the sum of the series of integers from 1 to n+1 where n = number of items to be distributed.
In this ques n = 5, thus the answer is 21.
For n = 6 it will be 28. For n = 7 it will be 36.....

However here is the explanation based on the counting method...

1) 1 person gets all 5 donuts -
Possibility: 3.

2). 2 persons get all 5 donuts -
The one without donuts - possibility - 3.
The other 2 persons have 4 ways.
Hence in total -- 3*4 = 12.

3). All have donuts -
The way to divide can only be - 1, 2, 2; 1, 3, 1
But this arrangement can get interchanged midst 3 persons, hence it becomes --
(3!)/ (2) + (3!)/ (2) = 6.

(3!)/ (2) is needed because donuts are all same without difference.
3! is counting the same arrangement twice.

Hence the answer - 3 + 12 + 6 = 21.