Question

The number of words that can be formed by using all the letters of the word 'MISSISSIPPI' is

• A. \( \frac{11!}{4!} \)
• B. \( 11! \)
• C. \( \frac{11!}{(4!)^2} \)
• D. \( \frac{11!}{(4!)^2 \cdot 2!} \)
  (E) \( \frac{11!}{(4!)^2 \cdot 2!} \)
• E. \( \frac{11!}{(4!)^2 \cdot 2!} \)

Hint:

Always double-check your counts! In 'MISSISSIPPI', it's 1M, 4I, 4S, 2P. The sum should equal the total length ($1+4+4+2 = 11$).

Answer 1

The Correct Option is D

Step 1: Understanding the Concept:
When arranging objects where some are identical, we divide the total number of permutations by the factorials of the counts of each repeating object to avoid overcounting.
Step 2: Key Formula or Approach:
Number of permutations = \( \frac{n!}{p! \cdot q! \cdot r! \dots} \), where $n$ is the total count and $p, q, r$ are the counts of repeating items.
Step 3: Detailed Explanation:
1. Count total letters in 'MISSISSIPPI': $n = 11$.
2. Count repetitions:
- M: 1 time
- I: 4 times
- S: 4 times
- P: 2 times
3. Apply the formula: \[ \text{Total arrangements} = \frac{11!}{4! (\text{for I}) \cdot 4! (\text{for S}) \cdot 2! (\text{for P})} \] \[ = \frac{11!}{(4!)^2 \cdot 2!} \]
Step 4: Final Answer:
The number of words is \( \frac{11!}{(4!)^2 \cdot 2!} \).