How many permutations can be made out of the letters of the word 'TRIANGLE ' ? How many of these will begin with T and end with E ?

Answer

To solve the problem of finding the number of permutations of the letters in the word "TRIANGLE" and how many of these permutations begin with 'T' and end with 'E', we can follow these steps:

### Step 1: Calculate the total number of permutations of the letters in "TRIANGLE".

The word "TRIANGLE" consists of 8 distinct letters. The formula for the number of permutations of n distinct objects is given by n!.

So, for the word "TRIANGLE":
\[ \text{Total permutations} = 8! = 8 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1 \]

Calculating this:
\[ 8! = 40320 \]

### Step 2: Calculate the number of permutations that begin with 'T' and end with 'E'.

If we fix 'T' at the beginning and 'E' at the end, we are left with the letters R, I, A, N, G, L (6 letters) to arrange in the middle.

The number of ways to arrange these 6 letters is given by:
\[ \text{Permutations of R, I, A, N, G, L} = 6! = 6 \times 5 \times 4 \times 3 \times 2 \times 1 \]

Calculating this:
\[ 6! = 720 \]

### Final Answers:
1. The total number of permutations of the letters in "TRIANGLE" is **40320**.
2. The number of permutations that begin with 'T' and end with 'E' is **720**.