Find the number of different words that can be formed from the letters of the word `TRIANGLE` so that no vowels are together.

Answer

To find 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 Vowels and Consonants
The word "TRIANGLE" consists of:
- Vowels: I, A, E (3 vowels)
- Consonants: T, R, N, G, L (5 consonants)

### Step 2: Arrange the Consonants
First, we arrange the consonants. The number of ways to arrange 5 consonants is given by:
\[ 5! = 120 \]

### Step 3: Determine the Gaps for Vowels
Once the consonants are arranged, we can visualize the arrangement as follows:
- For example, if the consonants are arranged as T R N G L, we can place gaps around them:
```
_ T _ R _ N _ G _ L _
```
This creates 6 gaps (one before each consonant and one after the last consonant) where we can place the vowels.

### Step 4: Choose Gaps for the Vowels
We need to select 3 out of these 6 gaps to place the vowels. The number of ways to choose 3 gaps from 6 is given by:
\[ \binom{6}{3} = 20 \]

### Step 5: Arrange the Vowels
The 3 vowels can be arranged among themselves in:
\[ 3! = 6 \]

### Step 6: Calculate the Total Arrangements
Now, we multiply the number of arrangements of consonants, the number of ways to choose gaps, and the arrangements of vowels:
\[ \text{Total arrangements} = (5!) \times \binom{6}{3} \times (3!) \]
Substituting the values we calculated:
\[ \text{Total arrangements} = 120 \times 20 \times 6 \]

### Step 7: Perform the Calculation
Calculating this gives:
\[ 120 \times 20 = 2400 \]
\[ 2400 \times 6 = 14400 \]

### Final Answer
Thus, the total number of different words that can be formed from the letters of the word "TRIANGLE" so that no vowels are together is:
\[ \boxed{14400} \]