In how many ways can the letters of the word "CAPITAL" be arranged such that the vowels always remain together?

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Accepted Answer

The word "CAPITAL" consists of 7 letters: C, A, P, I, T, A, L. Among these, the vowels are A, I, and A. To ensure that the vowels are always together, we treat them as a single unit or block (AIA).

### Step 1: Treating vowels as one unit  
Now, the letters to be arranged are: C, P, T, L, and (AIA). This gives us a total of **5 units to arrange**, which can be done in:  
$$
5! = 5 	imes 4 	imes 3 	imes 2 	imes 1 = 120 	ext{ ways.}
$$

### Step 2: Arranging the vowels within their group (AIA)  
Within the block (AIA), there are **3 letters**, where **A is repeated twice**. The number of distinct arrangements of these letters is:  
$$
\frac{3!}{2!} = \frac{3 	imes 2 	imes 1}{2 	imes 1} = 3 	ext{ ways.}
$$

### Step 3: Total arrangements  
The total number of arrangements where the vowels are together is:  
$$
120 	imes 3 = 360
$$

Thus, the total number of ways to arrange the letters of "CAPITAL" such that the vowels always come together is **360 ways**.

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