Number of words that can be made with the letters of the word `"GENIUS"` if each word neither begins with `G` nor ends in `S` is

Options

A. `24`
B. `240`
C. `480`
D. `504`

Explanation

To solve the problem of finding the number of words that can be formed with the letters of the word "GENIUS" such that no word begins with 'G' and does not end with 'S', we will follow these steps:

### Step 1: Identify the letters and their positions
The word "GENIUS" consists of 6 distinct letters: G, E, N, I, U, S.

### Step 2: Calculate the total arrangements without restrictions
The total number of arrangements of the letters in "GENIUS" is given by the factorial of the number of letters:
\[ 6! = 720 \]

### Step 3: Calculate arrangements that begin with 'G'
If a word begins with 'G', we have 5 remaining letters (E, N, I, U, S) to arrange. The number of arrangements is:
\[ 5! = 120 \]

### Step 4: Calculate arrangements that end with 'S'
If a word ends with 'S', we again have 5 remaining letters (G, E, N, I, U) to arrange. The number of arrangements is:
\[ 5! = 120 \]

### Step 5: Calculate arrangements that both begin with 'G' and end with 'S'
If a word begins with 'G' and ends with 'S', we have 4 remaining letters (E, N, I, U) to arrange. The number of arrangements is:
\[ 4! = 24 \]

### Step 6: Apply the principle of inclusion-exclusion
To find the total number of arrangements that either begin with 'G' or end with 'S', we use the inclusion-exclusion principle:
\[ \text{Total} = (\text{Begin with G}) + (\text{End with S}) - (\text{Begin with G and End with S}) \]
\[ \text{Total} = 120 + 120 - 24 = 216 \]

### Step 7: Calculate the valid arrangements
Now, subtract the total arrangements that begin with 'G' or end with 'S' from the total arrangements:
\[ \text{Valid arrangements} = \text{Total arrangements} - \text{Total} \]
\[ \text{Valid arrangements} = 720 - 216 = 504 \]

### Final Answer
The number of words that can be made with the letters of the word "GENIUS" that neither begin with 'G' nor end with 'S' is **504**.

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