The number of ways that the letters of the word 'corporation' be arranged so that the vowels always come together are c. 50, 400
How to find the number of ways?
Treat the vowels as a single unit: There are 3 consonants (C, R, P, N) and 1 group of vowels (AOI). We can arrange these 4 units in 4! ways.
Arrange the consonants within their unit: There are 3! ways to arrange the 3 consonants.
Arrange the vowels within their unit: There are 3! ways to arrange the 3 vowels (A, O, I).
Therefore, the total number of arrangements is:
= 4! x 3! x 3!
= 24 x 6 x 6
= 5040
Then:
= 5, 040 x 10
= 50,400 ways
In conclusion, there are 50,400 different ways to arrange the letters of 'corporation' so that the vowels always come together.