How many ways are there for arranging letters of the word AMAZING? How many ways are there for arranging letters of the word $AMAZING$ such that the '$I$' appears between the two '$A$'s ?

Answer given is $5$!, but I doubt that.
As a word $AMINZAG$ is also allowed here .
 A: If the letters $AIA$ must appear tight together you have that there is $5!$ ways. The reason is that you have the problem in permuting the fragments $AIA$, $M$, $Z$, $N$, $G$ that is $5$ fragments.
If on the other hand you only require that $I$ is somewhere between the $A$s, not requiring them to be tight together. This interpretation allows for $AMINZAG$ for example. The solution is to consider all the permutations of the letters and interpreting the $A$s to be distinct. This gives $7!$ permutations, but not all are allowed. You still have to consider the relative ordering of the $A$s and $I$, these can be ordered in $3!$ ways of which only $2$ are allowed (but they are also considered equal so while we multiply with $2$ we would also have to divide by $2$). The result taking this into account becomes:
$${7!\over 3!} = 840$$
Considering the given answer is $5!$ we therefore can conclude that it's the first interpretation that is the "correct".
A: Since you want $I$ between two $A$'s, make a seperate unit of $$\underbrace{\boxed{AIA}}_{\text{Single Unit}}$$
Now what you are left with is, arranging $\boxed{AIA} , M , Z, N, G$ . You can arrange these in $5!$ ways.
A: I think the question meant directly between two $A's$.If so, then
Here's a solution:
$4!$ ways of writing it like this $AIA****$
$4!$ ways of writing it like this $*AIA***$
$4!$ ways of writing it like this $**AIA**$
$4!$ ways of writing it like this $***AIA*$
$4!$ ways of writing it like this $****AIA$
Total ways =$$4!+4!+4!+4!+4!=5(4!)=5!$$
