The number of arrangenments which can be made using all the letters of the word LAUGH, if the vowels are adjacent, is

Options

A. 10
B. 24
C. 48
D. 120

Answer

To solve the problem of finding the number of arrangements of the letters in the word "LAUGH" with the condition that the vowels (A and U) are adjacent, we can follow these steps:

### Step 1: Group the Vowels
Since we want the vowels A and U to be adjacent, we can treat them as a single unit or group. Let's denote this group as (AU).

### Step 2: Identify the Groups
Now, we can list the groups we have:
1. (AU) - the group of vowels
2. L
3. G
4. H

This gives us a total of 4 groups to arrange: (AU), L, G, H.

### Step 3: Arrange the Groups
The number of ways to arrange these 4 groups is given by the factorial of the number of groups:
\[ 4! = 24 \]

### Step 4: Arrange the Vowels Within Their Group
Next, we need to consider the arrangements of the vowels within their group (AU). The vowels A and U can be arranged among themselves in:
\[ 2! = 2 \text{ ways} \]

### Step 5: Calculate the Total Arrangements
To find the total number of arrangements where the vowels are adjacent, we multiply the number of arrangements of the groups by the arrangements of the vowels:
\[ \text{Total arrangements} = 4! \times 2! = 24 \times 2 = 48 \]

### Final Answer
Thus, the total number of arrangements of the letters in the word "LAUGH" with the vowels A and U adjacent is **48**.

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