find the number of distinguishable permutations if the letters M I S S I S S I P P I

To find the number of distinguishable permutations of the letters in the word **MISSISSIPPI**, we use the formula for permutations of multiset objects:

\[
\frac{n!}{k_1! \cdot k_2! \cdot k_3! \dots k_r!}
\]

Where:
- \(n\) is the total number of letters,
- \(k_1, k_2, \dots, k_r\) are the frequencies of the repeated letters.

For **MISSISSIPPI**, the total number of letters is:

- \( M = 1 \)
- \( I = 4 \)
- \( S = 4 \)
- \( P = 2 \)

So, the total number of letters \(n = 11\), and we apply the formula:

\[
\frac{11!}{1! \cdot 4! \cdot 4! \cdot 2!}
\]

Let's calculate this step by step.

First, compute the factorials:

\[
11! = 39916800, \quad 1! = 1, \quad 4! = 24, \quad 2! = 2
\]

Now, plug them into the formula:

\[
\frac{39916800}{1 \cdot 24 \cdot 24 \cdot 2} = \frac{39916800}{1152} = 34650
\]

So, the number of distinguishable permutations of the letters in **MISSISSIPPI** is **34,650**.

Would you like a breakdown of the steps or any further details?

Here are five questions to deepen your understanding:
1. How does the formula for permutations of a multiset differ from regular permutations?
2. What is the importance of factorials in combinatorics?
3. Can you calculate the number of permutations for smaller words, like "BOOK"?
4. How does the presence of identical objects affect the permutation count?
5. What if there were 3 P's in MISSISSIPPI, how would the formula change?

**Tip:** Always remember that factorials grow very quickly, so it's important to simplify terms before performing large calculations!

Learn how to calculate the number of distinguishable permutations of the word 'MISSISSIPPI' using the permutation of multiset formula. Explore key concepts in combinatorics and factorials with a detailed breakdown of the process. This problem is perfect for students in Grades 9-12.

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