Question 1175422
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The vowels in the word are A, E, E, and I.<br>
You need to do the calculations as a series of different cases, depending on which vowels are the first and last letters.<br>
(1) The first and last letters are A and I.<br>
Those two can be in either of two orders; the remaining 10 letters include 3 R's, 2 S's, and 2 E's.  The number of arrangements for this case is<br>
{{{(2)(10!/((3!)(2!)(2!)))}}}<br>
(2) The first and last letters are A and E.<br>
Those two can be in either of two orders; the remaining 10 letters include 3 R's and 2 S's.  The number of arrangements for this case is<br>
{{{(2)(10!/((3!)(2!)))}}}<br>
(3) The first and last letters are I and E.<br>
This case will have the same number of arrangements as case (2).<br>
(4) The first and last letters are E and E.<br>
There is only one way to arrange those; the remaining 10 letters include 3 R's and 2 S's.  The number of arrangements for this case is<br>
{{{10!/((3!)(2!))}}}<br>
Add the numbers of arrangements for the four cases to get the final answer.<br>