In how many ways can the letters of the word CLASSROOM be arranged such that the new word does not begin or end with a vowel?

In how many ways can the letters of the word CLASSROOM be arranged such that the new word does not begin or end with a vowel?

Hi Chris   

How about I do it a different way and we will see if our answers are the same  

The number of arrangements in total, if there were no restrictions is

\(\frac{9!}{2!2!}\)

9!/(2!*2!) = 90720

​

How many ways can it begin with a vowel

If it starts with an O then 8!/2! = 20160 ways

If it starts with an A then 8!/(2!2!) = 10080ways

It is the same for ending in a vowel so that is

2*(20160+10080) = 60480

BUT

there is double counting happening here because is it begins and ends in a vowel it has been inclued both times.

SO how many start AND end in a vowel;?

Start with A end in O =  7!/2! = 2520 ways

Start with O end in A =   2520 ways

Start with O end in O = 2520 ways

That is    3*2520 = 7560 ways

So altogether it seems that   60480-7560 = 52920  either start or end with a vowel.

So the number that DO NOT start OR end in a vowel is    90720-52920 = 37800

YEP we can't both be wrong....   LOL 

See I have been studying very hard!!

I'll give his one a try

We have 6  ways to choose a consonant for the first position and 5 remaining ways to choose a consonant for the last position

And the other letters can be arranged in  7! ways

However........we must divide this by  [2! *2!]   to count only  the "identifiable" words  since we have repeated "O's"  and "S's"

So we have

[ 6 * 5  * 7! ] / [2! * 2!]   = 37800  arrangements

[ Can some other mathematician (Melody??)  check this  ??? ]

Mountains Out Of Mole Hills.........indeed.......LOL!!!!!!!

Thanks, Melody......