Find the Generating Function for the number of distributions

Find the Generating Function for the number of distributions of r Identical objects into 10 Different boxes with at most 4 objects in each box?

Solution

Given maximum number of objects in each box= 4

The probability will be either 0 objects or 1 object or 2 or 3 or atmost 4

Then The generating function for number of distributions of r identical objects in 1 box at most 4 objects will be

(x^0+ x^1+x^2+x^3+x^4) = 1+ x + x^2 + x^3 + x^4 .......equation 1

But the question asked is

generating function for number of distributions of r identical objects in 10 different boxes at most 4 objects

Since for each box equation 1 gives the function ,The required function for 10 boxes will be ( 1+  x + x^2 + x^3 + x^4)^10