What is the number of words formed from the letters of the word 'JOKE' so that the vowels and consonants alternate?

Options

A. 4
B. 8
C. 12
D. None of these

Answer

To find the number of words formed from the letters of the word 'JOKE' such that the vowels and consonants alternate, we can follow these steps:

### Step 1: Identify the vowels and consonants
The word 'JOKE' consists of:
- Vowels: O, E (2 vowels)
- Consonants: J, K (2 consonants)

### Step 2: Determine the arrangement pattern
Since we have 2 vowels and 2 consonants, we can have two possible arrangements:
1. Vowel-Consonant-Vowel-Consonant (V-C-V-C)
2. Consonant-Vowel-Consonant-Vowel (C-V-C-V)

### Step 3: Calculate arrangements for each pattern
**Pattern 1: V-C-V-C**
- We can arrange the vowels (O, E) in the 1st and 3rd positions. The number of ways to arrange 2 vowels is:
\[ 2! = 2 \text{ ways} \]
- We can arrange the consonants (J, K) in the 2nd and 4th positions. The number of ways to arrange 2 consonants is:
\[ 2! = 2 \text{ ways} \]
- Therefore, the total arrangements for this pattern is:
\[ 2! \times 2! = 2 \times 2 = 4 \text{ ways} \]

**Pattern 2: C-V-C-V**
- We can arrange the consonants (J, K) in the 1st and 3rd positions. The number of ways to arrange 2 consonants is:
\[ 2! = 2 \text{ ways} \]
- We can arrange the vowels (O, E) in the 2nd and 4th positions. The number of ways to arrange 2 vowels is:
\[ 2! = 2 \text{ ways} \]
- Therefore, the total arrangements for this pattern is:
\[ 2! \times 2! = 2 \times 2 = 4 \text{ ways} \]

### Step 4: Add the total arrangements from both patterns
Now, we add the total arrangements from both patterns:
\[ 4 \text{ (from V-C-V-C)} + 4 \text{ (from C-V-C-V)} = 8 \text{ ways} \]

### Final Answer
The total number of words formed from the letters of the word 'JOKE' such that the vowels and consonants alternate is **8**.

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