\problemdata      
        {counting-13} % latex-friendly label for the prob.
        {Counting} % The topic(s) of the problem content.
        {?} % Source (if known)
        {S02 CP8F; S99, Q2} % Usage (list ones you are aware of).
        {Christos Kapoutsis, S02} % Last revision info (author, date).


\begin{problem} The \emph{king positioning} in an
arrangement of a deck of 52 playing cards is the sequence of numerical
positions in the deck, from one to 52, of the four successive kings.  For
example, the king positioning $(1,2,3,4)$ means all the kings come at the
beginning of the deck.  The king positioning $(1,18,35,52)$ describes the
situation in which the kings are spaced uniformly---with exactly 16 cards
between successive kings.

\begin{problemparts}
\problempart How many king positionings are there?

\solution{$\dbinom{52}{4}$.}

\problempart How many king positionings are there in which \emph{no two
kings are adjacent}?

\solution{Think of kings as bars and the remaining 48 cards as stars.
Place one star between each two adjacent kings (bars).  This requires
three stars and leaves 45 stars that can be placed anywhere; there are
$\dbinom{45+4}{4}$ ways to do this.}

\problempart Of the $52!$ possible arrangements of the deck, how many
have no two kings adjacent?

\solution{Now in each of the previous patterns of stars and bars, the
kings can be permuted in $4!$ ways and the remaining cards can be permuted
in $48!$ ways, so the total number of desired arrangements is
\[
\binom{45+4}{4} 48!\, 4! = 49!\, 48!/45! = 48!\, P(49,4).
\]

\iffalse
The form of the RHS suggests that there is another way to do this count,
but I haven't had time to figure what it is.
\fi
}

\end{problemparts}
\end{problem}

