The total number of words that can be formed using all letters of the word 'RITESH' that neither begins with I nor ends with R, is

Options

A. 504
B. 480
C. 600
D. 720

Explanation

To solve the problem of finding the total number of words that can be formed using all letters of the word 'RITESH' that neither begins with 'I' nor ends with 'R', we can follow these steps:

### Step 1: Calculate the total number of arrangements of the letters in 'RITESH'.
The word 'RITESH' has 6 distinct letters. The total number of arrangements of these letters is given by:
\[ 6! = 720 \]

### Step 2: Calculate the number of arrangements that begin with 'I'.
If the first letter is fixed as 'I', we are left with the letters 'R', 'T', 'E', 'S', 'H' (5 letters). The number of arrangements of these 5 letters is:
\[ 5! = 120 \]

### Step 3: Calculate the number of arrangements that end with 'R'.
If the last letter is fixed as 'R', we are left with the letters 'I', 'T', 'E', 'S', 'H' (5 letters). The number of arrangements of these 5 letters is:
\[ 5! = 120 \]

### Step 4: Calculate the number of arrangements that begin with 'I' and end with 'R'.
If 'I' is the first letter and 'R' is the last letter, we are left with the letters 'T', 'E', 'S', 'H' (4 letters). The number of arrangements of these 4 letters is:
\[ 4! = 24 \]

### Step 5: Apply the principle of inclusion-exclusion.
To find the total number of arrangements that either begin with 'I' or end with 'R', we use the inclusion-exclusion principle:
\[ \text{Total (begin with I or end with R)} = (\text{begin with I}) + (\text{end with R}) - (\text{begin with I and end with R}) \]
Substituting the values we calculated:
\[ = 120 + 120 - 24 = 216 \]

### Step 6: Subtract from the total arrangements to find the desired count.
Now, we subtract the number of arrangements that begin with 'I' or end with 'R' from the total arrangements:
\[ \text{Desired arrangements} = \text{Total arrangements} - \text{Total (begin with I or end with R)} \]
\[ = 720 - 216 = 504 \]

Thus, the total number of words that can be formed using all letters of the word 'RITESH' that neither begins with 'I' nor ends with 'R' is **504**.

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