How many arrangements of the word marmalade can be made with the vowels in the original order How many arrangements of the word MARMALADE can be made with the vowels in the original order? What is the procedure for doing this problem? Is there more than one way of approaching it?
 A: We first change the problem a bit, and deal with the word marMalade.  
List the vowels in their original order: a a a e.  They determine $5$ "gaps" (we are counting the space before the first a and the space after the $e$ as gaps).
Now look at the letters m r M l d. The first of these, namely m, can be slipped into any of the $5$ gaps. After we do that we have a $5$-letter words, i.e. $6$ "gaps." The letter $r$ can be slipped into any of these $6$ gaps. 
Now we have a $6$-letter word, that is, $7$ gaps. We can slip $M$ into any of these. Continue. We end up with 
$5\cdot 6\cdot 7\cdot 8\cdot 9$ "words."
Now turn the M back into an m. This divides the number of words by $2$. Thus the total number of words for the original problem is
$$\frac{5\cdot 6\cdot 7\cdot 8\cdot 9}{2}.$$
There are many other ways to solve the problem. 
A: An approach would be to consider the vowels as all one letter, such as in
$$
\text{MARMALADA}
$$
where it is understood that we want to replace $\text{AAAA}\to\text{AAAE}$.
Thus, for each permutation we wish to count there are $4!$ permutations of the $\text{A}$s and $2!$ permutations of the $\text{M}$s that don't get counted to get the $9!$ total permutations of $9$ letters.
Thus, there are $\frac{9!}{4!2!}$ permutations of the letters in $\text{MARMALADA}$ which is the number of permutations of the letters in $\text{MARMALADE}$ that keep the vowels in order.
A: There will be more than one approach.  Here is one set of hints:


*

*Look at vowels: AAAE.  There are four.

*Look at consonants: MRMLD.  There are five including a pair identical.

*How many ways of arranging the consonants taking account of the fact there is a pair? 

*How many ways of interleaving four vowels and five consonants already in a given order?


Multiply those two together.
