# Math Help - permutations problem

1. ## permutations problem

from the word STUDIO, how many arrangements are possible if the letter O is not the first or the last?

2. Put the O in the first place and arrange the other 5

Put the O in the last place and arrange the other 5

Add them and subtract from the total number of arrangements.

3. Hello, Bartimaeus!

Another approach . . .

From the word STUDIO, how many arrangements are possible
if the letter O is not the first or the last?

If the O is not first or last, there are 4 places for it: . $X\;\_ \;\_ \;\_ \;\_ \;X$

Then the other 5 letters can be placed in: . ${\bf5!}$ ways.

Therefore, there are: . $4\cdot5! \:=\:480$ arrangements.

4. I got the same thing by doing it the way I mentioned.

Place the O on the end and arrange the other 5 in 120 ways.

Place the O in the front and arrange the other 5 in 120 ways.

That is 240 ways.

There are 6!=720 arrangements altogether.

So, 720-240=480 ways