How many ways are there to put 6 b***s in 3 boxes if the b***s are not distinguishable but the boxes are?

Let me answer the first question:  How many ways can you put 6 b***s in 3 boxes if the b***s are not distinguishable but the boxes are.

 

This can be answered by finding all the 3-digit numbers whose digits sum to 6. The first digit represents the number of b***s in the first box, the second digit represents the number of b***s in the second box, and the third digit represents the number of b***s in the third box.

 

The possible 3-digit numbrs are:

006     105     204     303     402     501     600

015     114     213     312     411     510

024     123     222     321     420

033     132     231     330

042     141     240

051     150

060

 

So there are 7 + 6 + 5 + 4 + 3 + 2 + 1  =  28 ways.