Words are formed with the letter of the word TRIANGLE. The ratio of the words starting with T and ending with E to the words ending with G is:

Options

A. `8:3`
B. `7:1`
C. `3:8`
D. `1:7`

Answer

To solve the problem of finding the ratio of the words starting with T and ending with E to the words ending with G using the letters of the word "TRIANGLE," we will follow these steps:

### Step 1: Identify the letters in the word "TRIANGLE"
The word "TRIANGLE" consists of 8 letters: T, R, I, A, N, G, L, E.

### Step 2: Calculate the number of words starting with T and ending with E
When we fix T at the beginning and E at the end, we have the following letters left to arrange: R, I, A, N, G, L (6 letters).

The number of ways to arrange these 6 letters is given by \(6!\) (6 factorial).

\[ 6! = 720 \]

### Step 3: Calculate the number of words ending with G
When we fix G at the end, we have the following letters left to arrange: T, R, I, A, N, L, E (7 letters).

The number of ways to arrange these 7 letters is given by \(7!\) (7 factorial).

\[ 7! = 5040 \]

### Step 4: Form the ratio
Now, we need to find the ratio of the number of words starting with T and ending with E to the number of words ending with G.

The ratio can be expressed as:

\[ \text{Ratio} = \frac{6!}{7!} \]

### Step 5: Simplify the ratio
Using the property of factorials, we know that \(7! = 7 \times 6!\). Thus, we can simplify the ratio:

\[ \text{Ratio} = \frac{6!}{7 \times 6!} = \frac{1}{7} \]

### Conclusion
The ratio of the words starting with T and ending with E to the words ending with G is:

\[ 1 : 7 \]