MHB Solving a Practice Exam Question: Arranging Letters in ROCKET

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The discussion centers on calculating the arrangements of the letters in "ROCKET" with the condition that the vowels E and O must stay together. The correct approach is to treat the vowels as a single unit, resulting in five units to arrange: R, OE, C, K, and T. This leads to the calculation of 5! for the arrangements of these units, while the factor of 2 accounts for the two possible orders of the vowels (EO or OE). The confusion arises from the assumption that only four letters are left after selecting the vowels, but the vowels are treated as one unit in this scenario. Understanding this concept clarifies the reasoning behind the calculation.
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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?
 
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nari said:
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.
 
Opalg said:
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?
 
nari said:
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?
 
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