The number of different words can be formed from the letters of the word TRIANGLE so that no vowels are together is

Options

A. 7200
B. 36000
C. 14400
D. 1240

Answer

To solve the problem of finding the number of different words that can be formed from the letters of the word "TRIANGLE" such that no vowels are together, we can follow these steps:

### Step 1: Identify the letters and their types
The word "TRIANGLE" consists of 8 letters:
- Vowels: I, A, E (3 vowels)
- Consonants: T, R, N, G, L (5 consonants)

### Step 2: Arrange the consonants
First, we will arrange the consonants. There are 5 consonants (T, R, N, G, L). The number of ways to arrange these consonants is given by the factorial of the number of consonants:

\[ \text{Arrangements of consonants} = 5! = 120 \]

### Step 3: Identify the gaps for placing vowels
Once we have arranged the consonants, we can visualize the arrangement as follows (C represents a consonant):

\[ C \_ C \_ C \_ C \_ C \]

This arrangement creates 6 gaps (represented by underscores) where we can place the vowels. The gaps are before the first consonant, between the consonants, and after the last consonant.

### Step 4: Choose gaps for vowels
We need to choose 3 gaps from the 6 available gaps to place the vowels. The number of ways to choose 3 gaps from 6 is given by the combination formula:

\[ \text{Ways to choose gaps} = \binom{6}{3} \]

Calculating this:

\[ \binom{6}{3} = \frac{6!}{3!(6-3)!} = \frac{6 \times 5 \times 4}{3 \times 2 \times 1} = 20 \]

### Step 5: Arrange the vowels
The vowels (I, A, E) can be arranged among themselves in:

\[ \text{Arrangements of vowels} = 3! = 6 \]

### Step 6: Calculate the total arrangements
Now, we can calculate the total number of arrangements where no vowels are together by multiplying the arrangements of consonants, the ways to choose gaps, and the arrangements of vowels:

\[ \text{Total arrangements} = (\text{Arrangements of consonants}) \times (\text{Ways to choose gaps}) \times (\text{Arrangements of vowels}) \]

Substituting the values we calculated:

\[ \text{Total arrangements} = 120 \times 20 \times 6 \]

Calculating this gives:

\[ \text{Total arrangements} = 120 \times 20 = 2400 \]
\[ 2400 \times 6 = 14400 \]

### Final Result
Thus, the number of different words that can be formed from the letters of the word "TRIANGLE" such that no vowels are together is **14400**.

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