How many different anagrams of...?

[Norm850]

How many different anagrams (including nonsensical words) can be made from FACETIOUSLY if we require that all six vowels must remain in alphabetical order (but not necessarily contiguous with each other)?

How do you do this? This homework is related to equivalence classes and partitions. I need to learn this, so don't just give the answer.

Thanks​

 

[soroban]

Hello, Norm850!

How many different anagrams (including nonsensical words) can be made from FACETIOUSLY if we require
that all six vowels must remain in alphabetical order (but not necessarily contiguous with each other)?

We have 11 spaces to fill.

Select 5 of the spaces for the vowels: .\(\displaystyle {11\choose5}\) choices.
Place the vowels in those spaces in alphabetical order.

Now place the 6 consonants in the remaining 6 spaces: .\(\displaystyle 6!\) choices.

Therefore, there are: .\(\displaystyle 462\cdot 720 \:=\:332,640\) anagrams.

 

[Norm850]

Hello, Norm850!


We have 11 spaces to fill.

Select 5 of the spaces for the vowels: .\(\displaystyle {11\choose5}\) choices.
Place the vowels in those spaces in alphabetical order.

Now place the 6 consonants in the remaining 6 spaces: .\(\displaystyle 6!\) choices.

Therefore, there are: .\(\displaystyle 462\cdot 720 \:=\:332,640\) anagrams.

They consider Y as a vowel. So I assume the correct answer is (11 choose 6)*5! => 11!/6! ?