I have the following math problem that I am stuck on:
How many words can be made from REGULATION where all the vowels are in alphabetical order?
I solved it two different ways and got two different answers.
1. I did 10C5 for choosing 5 spots then multiplied by 1 for the one order the vowels can be in. Then I multiplied by 5! for all the rearrangements of the other letters. This is (10 C 5) * 1 * 5! = 30240
2. I recognize the symmetry where half the arrangements, a is before e, half the arrangements e is before i, half the arrangements i is before o, half the arrangements o is before u. This symmetry means that the number of arrangements is 10!/(2^4)
which equals 226800.
Which is correct? Why is the other wrong?
Thanks for the help