In how many ways the letters of the word AFLATOON be arranged if the consonants and vowels must occupy alternate places?

Options

A. a. 144
B. b. 143
C. c. 288
D. d. 248

Answer

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

### Step 1: Identify the letters and their counts
The word "AFLATOON" consists of 8 letters:
- Vowels: A, A, O, O (4 vowels)
- Consonants: F, L, T, N (4 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 arrangements:
1. Vowel in the first position: V C V C V C V C
2. Consonant in the first position: C V C V C V C V

### Step 3: Calculate arrangements for vowels
In the arrangement, we have 4 vowels (A, A, O, O). The number of ways to arrange these vowels, considering the repetitions, is given by the formula for permutations of multiset:
\[ \text{Arrangements of vowels} = \frac{4!}{2! \times 2!} = \frac{24}{4} = 6 \]

### Step 4: Calculate arrangements for consonants
Next, we arrange the 4 consonants (F, L, T, N). Since all consonants are unique, the number of arrangements is:
\[ \text{Arrangements of consonants} = 4! = 24 \]

### Step 5: Calculate total arrangements for the first pattern (V C V C V C V C)
For the arrangement starting with a vowel:
\[ \text{Total arrangements} = \text{Arrangements of vowels} \times \text{Arrangements of consonants} = 6 \times 24 = 144 \]

### Step 6: Calculate total arrangements for the second pattern (C V C V C V C V)
For the arrangement starting with a consonant, the calculations are the same:
\[ \text{Total arrangements} = \text{Arrangements of consonants} \times \text{Arrangements of vowels} = 24 \times 6 = 144 \]

### Step 7: Combine both arrangements
Finally, we add the total arrangements from both patterns:
\[ \text{Total arrangements} = 144 + 144 = 288 \]

### Final Answer
Thus, the total number of ways to arrange the letters of the word "AFLATOON" such that consonants and vowels occupy alternate places is **288**.

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