Math Is Fun Forum

The Rosen's book, problem 5.5 #50:
How many ways are there to distribute 5 distinguishable objects into 3 indistinguishable bins?

One approach to such problems I know is to imagine distinguishable objects as a long box with compartments, indistinguishable objects as balls and by looking at the balls and separators between compartments we are getting a string of "O" and "l" which can be permuted and that would be the answer...
The only problem here is that as far as I understand this approach, it works perfectly well if number of indistinguishable objects are greater than number of distinguishable.  But if number of distinguishable is greater (like in the problem) I am getting strings like:

a, b, c, d, e here are distinguishable objects represented as compartments. But with this representation I always have some empty compartments - distinguishable objects which are not in any indistinguishable bins...

Another, more promising, approach is to just make a list from letters and separators: "abcde||", "abcd|e|", etc. That would accurately represent distinguishable objects and distinguishable bins. But the original problem states that bins should be indistinguishable, so "abcde||" == "|abcde|" == "||abcde"...
.....hmm....
The process of writing this topic gave me an idea. If we define objects as O and bins as B, then to get the answer it should be enough to count the number of permutations of the string (

) and divide them by number of bins. Am I right or not?

I spent a week thinking about this problem and now I can not believe the answer can be so simple...

I am working Rosen's book. Problem 4.1#6:
Prove that

whenever

I am supposed to do it by using induction, so my basis step is:

And the inductive step is (for n+1):

And now I am completely stuck.
Please, point me to the right direction on how to simplify those factorials?