The number of words that can be formed by using all the letters of the word 'PROBLEM' only once is :

Options

A. 5!
B. 6!
C. 7!
D. 8!

Answer

To find the number of distinct words that can be formed using all the letters of the word "PROBLEM," we can follow these steps:

### Step 1: Count the number of letters in the word
The word "PROBLEM" consists of 7 letters: P, R, O, B, L, E, M.

### Step 2: Check for repeated letters
In the word "PROBLEM," all the letters are distinct. This means there are no repeated letters.

### Step 3: Use the formula for permutations
Since all the letters are distinct, the number of ways to arrange these letters is given by the factorial of the number of letters. The formula for the number of permutations of n distinct objects is n!.

### Step 4: Calculate the factorial
Here, n = 7 (since there are 7 letters). Therefore, we need to calculate 7! (7 factorial).

7! = 7 × 6 × 5 × 4 × 3 × 2 × 1 = 5040

### Conclusion
The total number of distinct words that can be formed by using all the letters of the word "PROBLEM" is 5040.

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