Find the number of permutations of the letters of the word 'ENGLISH'. How many of these begin with E and end with I?

Answer

To solve the problem, we need to find the number of permutations of the letters in the word "ENGLISH" and then determine how many of those permutations begin with 'E' and end with 'I'.

### Step 1: Count the total number of letters in "ENGLISH".
The word "ENGLISH" consists of 7 distinct letters: E, N, G, L, I, S, H.

### Step 2: Calculate the total number of permutations.
Since all the letters are distinct, the total number of permutations can be calculated using the factorial of the number of letters.

\[ \text{Total permutations} = 7! = 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1 = 5040 \]

### Step 3: Find the number of permutations that begin with 'E' and end with 'I'.
When we fix 'E' at the beginning and 'I' at the end, we are left with the letters N, G, L, S, which are 5 letters.

### Step 4: Calculate the permutations of the remaining letters.
The number of ways to arrange the 5 remaining letters (N, G, L, S) is given by the factorial of the number of letters left.

\[ \text{Permutations starting with E and ending with I} = 5! = 5 \times 4 \times 3 \times 2 \times 1 = 120 \]

### Final Answers:
1. The total number of permutations of the letters in "ENGLISH" is **5040**.
2. The number of permutations that begin with 'E' and end with 'I' is **120**.

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