Alphabetical arrangement of 5 letters - MUSIC

Question

"If you arrange the letters , U, S, M, I, C in dictionary order, in which position is the arrangement MUSIC"?

The answer I've worked out is 72 (working out below), but I just wanted to verify this with someone to see if my logic was correct!

How I worked it out:

There are 5! permutations of the letters. 24 of these start with the letter C, 24 with the letter I and 24 with the letter M (etc.)

So, already, there are 48 'words' before MUSIC. Now we start listing the permutations starting with the letter C:

MCISU MCIUS MCSIU MCSUI MCUIS MCUSI

and so on.

Thus there are also 6 arrangements starting with 'MI...' and six arrangements starting with 'MS...'.

So far, we're on 48+18 = 66 words

Now we start listing the arrangements starting 'MU...':

MUCIS MUCSI MUICS MUISC MUSCI MUSIC

And so, we can see that 'MUSIC' is the 72nd word when the words are arranged in alphabetical order.

Is this correct?

Answer 1

Your answer is correct but you could simplify this a bit -

Letters starting with either $C$ or $I$ or $M = 3 \times 4! = 72$

In $MUSIC$ - leaving $M$, all other letters are in perfect descending order so it has to be the last word starting with $M$ and hence it is in $72$nd position.