Hello,

Here is the major difference between identical-looking marbles into distinct plates  AND into identical-looking plates. What is most important is that we begin by identical-looking plates (by ''locking out the order''.) Here is a good example for you to see. Let's consider a+b+c=5 where a,b,c are whole numbers. When you perform partition of natural numbers, which tell us about the number of representing cases, we lock out the order between a, b, and c. Assume a \geq b \geq c . Then, 

a+b+c=5

5+0+0=5

4+1+0=5

3+2+0=5

3+1+1=5

2+2+1=5

Here is a basic summary : 

Identical-looking marbles into identical-looking plates = partition of natural numbers = locking the order between the plates (since we don't want order). 

No order means one specific order, and this order is produced by locking the order between the plates. Now, what is combination allowing repetitions? We put order back into plates. Let's bring our example back!

a+b+c=5

5+0+0=5 --> (5,0,0), (0,5,0), (0,0,5)

4+1+0=5 --> (4,1,0), (4,0,1), (1,0,4), (1,4,0), (0,4,1), (0,1,4)

3+2+0=5 --> (3,2,0), (3,0,2), (2,3,0), (2,0,3), (0,3,2), (0,2,3)

3+1+1=5 --> (3,1,1), (1,3,1), (1,1,3)

2+2+1=5 --> (2,2,1), (2,1,2), (1,2,2)

Do you notice that I am bringing back order among a, b, and c? This is combination allowing repetitions. We first perform natural partition and put back orders, when applying combination allowing repetitions!

Yay!! Thanks for listening to my lecture. Ask me any question that bothers you! :)