Final answer:
The number of arrangements of the word 'university' that contain the word 'rest' and have the vowels in alphabetical order is 48.
Step-by-step explanation:
To find the number of arrangements of the word 'university' that contain the word 'rest' and have the vowels in alphabetical order, we need to break down the problem. First, we need to find the number of ways to arrange the letters in 'rest' within the word 'university'.
'Rest' is a 4-letter word, so it has 4! = 24 arrangements. Next, we need to find the number of ways to arrange the remaining letters in 'university', considering that the vowels must be in alphabetical order. The remaining letters are 'univity', and the vowels in alphabetical order are 'ei'.
There are 2 vowels in 'univity', so there are 2! = 2 ways to arrange them. Therefore, the total number of arrangements of 'university' that contain the word 'rest' and have the vowels in alphabetical order is 24 x 2 = 48.