In how many different ways can the letters of the word 'SALOON' be arranged
If the consonants and vowels must occupy alternate places?

Answer

To solve the problem of arranging the letters of the word 'SALOON' such that consonants and vowels occupy alternate places, we can follow these steps:

### Step 1: Identify the letters in the word 'SALOON'
The word 'SALOON' consists of:
- Vowels: A, O, O (3 vowels)
- Consonants: S, L, N (3 consonants)

### Step 2: Determine the arrangement pattern
Since we need to arrange the letters such that consonants and vowels occupy alternate places, we can have two possible patterns:
1. Vowel - Consonant - Vowel - Consonant - Vowel - Consonant
2. Consonant - Vowel - Consonant - Vowel - Consonant - Vowel

### Step 3: Calculate arrangements for the first pattern (Vowel - Consonant)
- **Arranging the vowels (A, O, O)**:
The number of ways to arrange the vowels is given by the formula for permutations of multiset:
\[ \text{Arrangements of vowels} = \frac{3!}{2!} = \frac{6}{2} = 3 \]
(Here, 3! is for the total arrangements of the vowels, and 2! accounts for the repetition of 'O').

- **Arranging the consonants (S, L, N)**:
The number of ways to arrange the consonants is:
\[ \text{Arrangements of consonants} = 3! = 6 \]

### Step 4: Calculate total arrangements for the first pattern
The total arrangements for the first pattern (Vowel - Consonant) is:
\[ \text{Total arrangements for pattern 1} = \text{Arrangements of vowels} \times \text{Arrangements of consonants} = 3 \times 6 = 18 \]

### Step 5: Calculate arrangements for the second pattern (Consonant - Vowel)
Using the same reasoning as above:
- **Arranging the consonants (S, L, N)**:
\[ \text{Arrangements of consonants} = 3! = 6 \]

- **Arranging the vowels (A, O, O)**:
\[ \text{Arrangements of vowels} = \frac{3!}{2!} = 3 \]

### Step 6: Calculate total arrangements for the second pattern
The total arrangements for the second pattern (Consonant - Vowel) is:
\[ \text{Total arrangements for pattern 2} = \text{Arrangements of consonants} \times \text{Arrangements of vowels} = 6 \times 3 = 18 \]

### Step 7: Combine the arrangements from both patterns
Finally, we add the total arrangements from both patterns:
\[ \text{Total arrangements} = 18 + 18 = 36 \]

Thus, the total number of different ways the letters of the word 'SALOON' can be arranged such that consonants and vowels occupy alternate places is **36**.

---