Question 1091943
<br>Sorry to confuse you with (up to now) three different answers to your question....<br>
You are asking for the number of different arrangements of
{{{AAAABBCC}}}<br>
By a well known counting principle, the number of distinct arrangements is 
{{{(8!)/((4!)(2!)(2!)) = 420}}}<br>
If you aren't familiar with that counting principle, here is a quick explanation....<br>
If all eight items were different, then the number of arrangements would clearly be
{{{N = 8!}}}<br>
Because 4 items are the same, they can be arranged in 4!=24 different ways that are indistinguishable.  So we need to divide the number of distinct arrangements by 4!.
{{{N = (8!)/(4!)}}}<br>
Because 2 other items are the same, they can be arranged in 2!=2 different ways that are indistinguishable.  So we need to further divide the number of distinct arrangements by 2!.
{{{N = (8!)/((4!)(2!))}}}<br>
Finally, because 2 other items are the same, they can be arranged in 2!=2 different ways that are indistinguishable.  So we need to further divide the number of distinct arrangements again by 2!.
{{{N = (8!)/((4!)(2!)(2!))}}}<br>
For another example of the concept, the number of distinct arrangements of the letters MISSISSIPPI (11 letters; 4 the same, 4 others the same, 2 others the same) is
{{{(11!)/((4!)(4!)(2!))}}}<br>
Or...<br>
The number of distinct orders in which you can get 5 heads and 3 tails when you flip a coin 8 times is
{{{(8!)/((5!)(3!))}}}