\problemdata      
        {counting-8} % latex-friendly label for the prob.
        {Counting} % The topic(s) of the problem content.
        {Radhika Nagpal} % Source (if known)
        {S02 CP8W} % Usage (list ones you are aware of).
        {Christos Kapoutsis, S02} % Last revision info (author, date).



\begin{problem} 
\begin{problemparts}
\problempart How many different permutations are there of the word
MISSISSIPPI ?

\solution{
\[
\dbinom{11}{1,4,4,2} =\frac{11!}{1!4!4!2!} = 34650
\]
}

\problempart Suppose we have a generalized $n$-letter ``MISSISSIPPI''
word whose letters break down as follows: $m$ occurrences of ``M'',
$i$ occurrences of ``I'', $s$ of ``S'', and $p$ of ``P''.  For
example, for ``MISSISSIPI'' from the previous part, $(n,m,i,s,p) =
(11,1,4,4,2)$.

Identify which formulas among the following describe the number of
different permutations of the generalized ``MISSISSIPI'' word:

\begin{enumerate}

\item	$\dbinom{n}{m,i,s,p}$

\solution{True. This is a straight forward application of Theorem~6.2
in Course Notes~8.}

\item	$\dbinom{n}{p,i,m,s}$

\solution{True. The order of the lower arguments of the multinomial
coefficient is not important.}

\item	$\dbinom{n}{m,i,s,n-m-i-s}$

\solution{True. Because $m+i+s+p=n$, so $p=n-m-i-s$.}

\item	$\dbinom{n}{m}\dbinom{n}{i}\dbinom{n}{s}\dbinom{n}{p}$

\solution{False.}

\item	$\dbinom{n}{m}\dbinom{m}{i}\dbinom{i}{s}\dbinom{s}{p}$

\solution{False.}

\item   $\dbinom{n}{m}\dbinom{n-m}{i}\dbinom{n-m-i}{s}\dbinom{n-m-i-s}{p}$

\solution{True. With the same reasoning as the one used in the
alternative proof of Theorem~6.2, in Course Notes~8,~\S6.3.}

\end{enumerate}
\end{problemparts}

{\em Reminder:}
$\binom{r_1 + r_2 + \dots + r_n}{r_1, r_2, \dots, r_n}
= 
\frac{(r_1 + r_2 + \dots + r_n)!}{r_1! \ r_2! \ \dots \ r_n!}$

\end{problem}
