\problemdata      
        {?} % latex-friendly label for the prob.
        {Counting} % The topic(s) of the problem content.
        {?} % Source (if known)
        {Fall 2001, quiz 3} % Usage (list ones you are aware of).
        {Christos Kapoutsis, S02} % Last revision info (author, date).


\begin{problem}
A juggler plans to juggle 15 rings for his final\'e---more than any other
circus performer has ever managed.  In his bag are 9 red rings, 11 blue
rings, 13 white rings, and 15 rings striped with all three colors.  He
must select at least two of each kind of ring to be patriotic.  An example
of a patriotic choice of 15 rings would be: 2 red, 6 white, 3 blue and 4
striped.  How many different patriotic choices are there for the 15 rings?

\solution{$120$.

This is a ``stars and bars'' problem: the same as asking how many ways
there are to arrange 15 stars (the rings) and three bars such that there
are at least two stars in each of the four groups (the kinds of rings)
delimited by the bars.  Reserving eight stars for this purpose leaves 7
stars, so there are
\[
\binom{3 + 7}{3} = \binom{10}{3}.
\]
Note that the initial number of each kind of ring available to
the juggler is irrelevant as long as there are at least 9 of each.
}
\end{problem}