In how many different ways can the letters of the word ‘CORPORATION’ be arranged so that the vowels always come together?
Remember:
-
Group the vowels together as a single “block” or unit.
(Treat the vowels as one entity so they always stick together.) -
Count the remaining units:
The number of units to arrange = Total units (consonants + vowel block). -
Arrange the units:
Find the number of ways to arrange the total units.
For example, with 7 units, it’s 7!7!7!. -
Arrange the vowels within the block:
If vowels repeat (like “O, O, O”), use the formula 5!3!\frac{5!}{3!}3!5! for repeated vowels. -
Multiply the two results:
Total=Arrange the units×Arrange the vowels inside the block\text{Total} = \text{Arrange the units} \times \text{Arrange the vowels inside the block}Total=Arrange the units×Arrange the vowels inside the block
