Counting problem with multiple urns and indistinguishable balls

Question

I have the following problem:

There are $4$ urns and each urn can only contain $10$ balls. Of how many ways can we place $10$ white balls in the $4$ urns? There are no restriction.

I did some examples trying to understand this. I did an example in wich I had $4$ urns, each urn had $2$ spaces, and I wanted to place $2$ balls in the urns. I found that I could place $2$ balls in $10$ ways.

Advanced Greetings.

Answer 1

It is a classic Stars and Bars question

Then you have 4 urns and 10 balls

example : wwww.www..www , w being the balls and the dots the urns.

Then apply the theorem 2 : number of ways $ = {\tbinom {n+k-1}{k-1}}={\tbinom {n+k-1}{n}}$ with n=10 and k=4 $ = {\tbinom {13}{3}}= 286 $

For the 2 balls case, n=2 and k=4 the result is $ = {\tbinom {5}{3}}= 10 $ . Your enumeration was correct.