Question

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

• A. \( 5! \)
• B. \( 6! \)
• C. \( 7! \)
• D. \( 8! \)
• E. \( 9! \)

Hint:

If a word contains repeated letters, you must divide by the factorial of the frequency of each repeating letter. For example, in the word "APPLE", the number of words is \( 5!/2! \) because 'P' repeats twice.

Answer 1

The Correct Option is C

Concept: The number of permutations of \( n \) distinct objects taken all at a time is given by \( n! \) (n factorial). A "word" in this context refers to any arrangement of the given letters, whether it has a dictionary meaning or not.

Step 1: Count the number of distinct letters in the word.
The word provided is "PROBLEM". The letters are: P, R, O, B, L, E, M. There are a total of 7 distinct letters.

Step 2: Calculate the total number of arrangements.
Since all 7 letters are unique and we must use each letter exactly once, the total number of different words that can be formed is the number of permutations of 7 distinct items: \[ \text{Total Words} = 7! \] Calculation: \( 7! = 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1 = 5040 \).