Question

How many ways are there to split four red, five blue, and seven black balls among (a) two boxes without restriction? (b) two boxes with no box empty?

Answer 1

For this question each arrangement  will match to a triple non-negative integers.

For RED BLUE AND BLACK RESPECTIVELY which are in the BOX A.

Now after this we will count the balls with the triples.

There are total of 5 ways in which the total of 4 objects can be placed in the 2 boxes

And 6 ways of putting 5 objects in the 2 boxes

Similarly 8 ways in which the 6 objects can be placed in the 2 boxes.

In this situation we wil make the equation like this:

5*6*8=240

and in the situation where the boxes are not distinct we will divide it with 2

240/2

Now there are only 2 ways in which we can have the box empty

in which either box 1 is empty or box 2 is empty

in that situation we will get 240-2 = 238

and if the boxes are not distinct then we get 120-1 = 119