Find the number of ways in which the letters of word 'MEDICAL' be arranged if A and E are together but all the vowels never come together.

Explanation

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 break down the solution into several steps.

### Step 1: Count the total arrangements with A and E together
1. Treat A and E as a single unit or block. So, we can represent the letters as: (AE), M, D, I, C, L.
2. This gives us a total of 6 units to arrange: (AE), M, D, I, C, L.
3. The number of ways to arrange these 6 units is given by \(6!\).
4. Since A and E can be arranged among themselves in \(2!\) ways, we multiply by this factor.

**Calculation:**
\[ 6! = 720 \]
\[ 2! = 2 \]
\[ \text{Total arrangements with A and E together} = 6! \times 2! = 720 \times 2 = 1440 \]

### Step 2: Count the arrangements where all vowels (A, E, I) are together
1. Now, treat A, E, and I as a single block. So, we can represent the letters as: (AEI), M, D, C, L.
2. This gives us a total of 5 units to arrange: (AEI), M, D, C, L.
3. The number of ways to arrange these 5 units is given by \(5!\).
4. The vowels A, E, and I can be arranged among themselves in \(3!\) ways.

**Calculation:**
\[ 5! = 120 \]
\[ 3! = 6 \]
\[ \text{Total arrangements with A, E, and I together} = 5! \times 3! = 120 \times 6 = 720 \]

### Step 3: Calculate the arrangements where A and E are together but all vowels are not together
1. To find the required arrangements, we subtract the arrangements where all vowels are together from the arrangements where A and E are together.

**Calculation:**
\[ \text{Required arrangements} = \text{Total arrangements with A and E together} - \text{Total arrangements with all vowels together} \]
\[ = 1440 - 720 = 720 \]

### 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**.

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