Understanding the Problem
To find the number of arrangements of the word "OCCUR" where the vowels are together, we first identify the vowels and consonants in the word.
- Vowels: O, U
- Consonants: C, C, R
Grouping the Vowels
We can treat the two vowels (OU) as a single unit. Thus, we consider the arrangement of the following units:
- (OU), C, C, R
This gives us a total of four units to arrange: (OU), C, C, R.
Calculating Arrangements
The total arrangements of these four units can be calculated using the formula for permutations of multiset:
- Total units = 4 (OU, C, C, R)
- Since "C" is repeated, we divide by the factorial of the count of "C".
Total arrangements = 4! / 2! = 24 / 2 = 12.
Arranging the Vowels
The vowels (O, U) can be arranged among themselves in:
- 2! = 2 ways (OU or UO).
Total Arrangements
Now, to find the total arrangements where vowels are together, we multiply the arrangements of the units by the arrangements of the vowels:
- Total = 12 (arrangements of units) * 2 (arrangements of vowels) = 24.
Final Answer
Thus, the number of different ways to arrange the word "OCCUR" such that the vowels come together is 24.