in how many ways can the letters of the word MOBILE be arranged so that at least two constonants remain together ?

Options

A. A. 2880
B. B. 3200
C. C. 576
D. D. 1600

Answer

To solve the problem of arranging the letters of the word "MOBILE" such that at least two consonants remain together, we can follow these steps:

### Step 1: Identify the letters in the word "MOBILE"
The word "MOBILE" consists of 6 letters: M, O, B, I, L, E. Among these:
- Consonants: M, B, L (3 consonants)
- Vowels: O, I, E (3 vowels)

### Step 2: Calculate the total arrangements of the letters
The total number of arrangements of the letters in "MOBILE" can be calculated using the factorial of the number of letters:
\[ \text{Total arrangements} = 6! = 720 \]

### Step 3: Calculate the arrangements where no two consonants are together
To find the arrangements where at least two consonants are together, we can use the complementary counting method. First, we calculate the arrangements where no two consonants are together.

1. **Arrange the vowels**: We first arrange the vowels O, I, E. The number of arrangements of the vowels is:
\[ 3! = 6 \]

2. **Place the consonants**: Once the vowels are arranged, we can place the consonants in the gaps created by the vowels. For the arrangement of 3 vowels, there are 4 possible gaps (before the first vowel, between the vowels, and after the last vowel):
- _ O _ I _ E _

We need to choose 3 out of these 4 gaps to place the consonants. The number of ways to choose 3 gaps from 4 is given by:
\[ \binom{4}{3} = 4 \]

3. **Arrange the consonants**: The number of arrangements of the consonants M, B, L is:
\[ 3! = 6 \]

4. **Total arrangements with no two consonants together**: Now, we multiply the arrangements of vowels, the ways to choose gaps, and the arrangements of consonants:
\[ \text{Arrangements with no two consonants together} = 3! \times \binom{4}{3} \times 3! = 6 \times 4 \times 6 = 144 \]

### Step 4: Calculate the arrangements where at least two consonants are together
Now, we can find the arrangements where at least two consonants are together by subtracting the arrangements where no two consonants are together from the total arrangements:
\[ \text{Arrangements with at least two consonants together} = \text{Total arrangements} - \text{Arrangements with no two consonants together} \]
\[ = 720 - 144 = 576 \]

### Final Answer
Thus, the number of ways the letters of the word "MOBILE" can be arranged such that at least two consonants remain together is **576**.

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