To solve the problem of arranging the letters of the word "MEDICAL" such that A and E are together, but all the vowels never come together, we can follow these steps:
### Step 1: Count the total letters and identify vowels
The word "MEDICAL" has 7 letters: M, E, D, I, C, A, L. The vowels in this word are A, E, and I.
### Step 2: Treat A and E as a single unit
Since A and E must be together, we can treat them as a single unit or box. Thus, we can represent the letters as:
- (AE), M, D, I, C, L
This gives us a total of 6 units to arrange: (AE), M, D, I, C, L.
### Step 3: Calculate arrangements with A and E together
The number of arrangements of these 6 units is given by 6! (factorial of 6). Additionally, A and E can be arranged within their box in 2! ways (AE or EA).
So, the total arrangements with A and E together is:
\[
6! \times 2! = 720 \times 2 = 1440
\]
### Step 4: Count arrangements where all vowels are together
Now, we need to consider the case where all vowels (A, E, I) are together. We can treat A, E, and I as a single unit or box as well. Thus, we can represent the letters as:
- (AEI), M, D, C, L
This gives us a total of 5 units to arrange: (AEI), M, D, C, L.
The number of arrangements of these 5 units is given by 5!. Additionally, A, E, and I can be arranged within their box in 3! ways (AEI, AIE, EAI, EIA, IAE, IEA).
So, the total arrangements with all vowels together is:
\[
5! \times 3! = 120 \times 6 = 720
\]
### Step 5: Subtract the arrangements where all vowels are together from those where A and E are together
To find the arrangements where A and E are together but all vowels are not together, we subtract the arrangements where all vowels are together from the arrangements where A and E are together:
\[
\text{Total arrangements with A and E together but not all vowels together} = 1440 - 720 = 720
\]
### Final Answer
The number of ways in which the letters of the word "MEDICAL" can be arranged such that A and E are together but all the vowels never come together is **720**.
