MHB Solving a Practice Exam Question: Arranging Letters in ROCKET

[nari]

Hi! There is this question in my practice exam:

How many ways can all the letters in the word ROCKET be arranged so that the vowels are always together?

The answer is 5! x 2.

I understand where the 2 is coming from. What I don't understand is the 5. Shouldn't it be 4!, since we already selected two out of six letters?

 

[Opalg]

Hi! There is this question in my practice exam:

How many ways can all the letters in the word ROCKET be arranged so that the vowels are always together?

The answer is 5! x 2.

I understand where the 2 is coming from. What I don't understand is the 5. Shouldn't it be 4!, since we already selected two out of six letters?

Think of the E and the O as being glued together. They then count as a single "letter", and you have 5! ways of arranging the five letters R OE C K and T. The 2 just tells you whether the E comes before or after the O.

 

[nari]

Think of the E and the O as being glued together. They then count as a single "letter", and you have 5! ways of arranging the five letters R OE C K and T. The 2 just tells you whether the E comes before or after the O.

So then the options would be: eo/oe + r o/e c k t?

 

[Jameson]

So then the options would be: eo/oe + r o/e c k t?

I don't quite follow the whole logic here but the beginning part is true, the letter pairs are "eo" or "oe". For simplicity let's just called these two letters P (for pair). Opalg suggested the same thing but maybe seeing it like this will help.

How many ways can you arrange RPCKT?