The number of words which can be formed from the letters of the word "MAXIMUM" if two consonants cannot occur together?

Options

A. 4!
B. `3!xx4!`
C. `3!`
D. `(4!)/(3!)`

Answer

To solve the problem of finding the number of words that can be formed from the letters of the word "MAXIMUM" such that no two consonants occur together, we can follow these steps:

### Step 1: Identify the letters and their types
The word "MAXIMUM" consists of:
- Vowels: A, I, U (3 vowels)
- Consonants: M, M, M, X (4 consonants)

### Step 2: Determine the arrangement of vowels
Since we want to ensure that no two consonants are together, we will first arrange the vowels. The vowels can be arranged in the even positions of the word.

### Step 3: Identify positions for vowels and consonants
The total positions available for the letters in the word "MAXIMUM" are 7 (since it has 7 letters). The positions can be numbered as follows:
1. 1 (odd)
2. 2 (even)
3. 3 (odd)
4. 4 (even)
5. 5 (odd)
6. 6 (even)
7. 7 (odd)

We will place the vowels in the even positions (2, 4, 6) and the consonants in the odd positions (1, 3, 5, 7).

### Step 4: Arrange the vowels
The vowels A, I, U can be arranged in the 3 even positions (2, 4, 6). The number of ways to arrange 3 vowels is given by:
\[ 3! = 6 \text{ ways} \]

### Step 5: Arrange the consonants
The consonants are M, M, M, X. We need to arrange these 4 consonants in the 4 odd positions (1, 3, 5, 7). The number of ways to arrange these consonants, taking into account the repetition of M, is given by:
\[ \frac{4!}{3!} = \frac{24}{6} = 4 \text{ ways} \]

### Step 6: Calculate the total arrangements
The total number of arrangements of the letters in the word "MAXIMUM" such that no two consonants are together is the product of the arrangements of the vowels and the arrangements of the consonants:
\[ \text{Total arrangements} = (3!) \times \left(\frac{4!}{3!}\right) = 6 \times 4 = 24 \]

### Final Answer
Thus, the total number of words that can be formed from the letters of the word "MAXIMUM" such that no two consonants occur together is **24**.

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