Determine the number of permutations of the letters in the words: phillipines

To determine the number of **permutations** of the letters in the word **"phillipines"**, we first need to observe if there are any repeated letters and account for them in the calculation.

### Step 1: Count the total number of letters.
The word **"phillipines"** contains 11 letters in total.

### Step 2: Count the frequency of repeated letters.
- **p** appears twice.
- **h** appears once.
- **i** appears three times.
- **l** appears twice.
- **n** appears once.
- **e** appears once.
- **s** appears once.

### Step 3: Use the formula for permutations of multiset.
For a word with repeated letters, the number of permutations is given by:

\[
\text{Permutations} = \frac{n!}{k_1! \times k_2! \times \cdots \times k_m!}
\]

where:
- \(n\) is the total number of letters (11),
- \(k_1, k_2, \ldots, k_m\) are the frequencies of the repeated letters.

Substitute the values:

\[
\text{Permutations} = \frac{11!}{2! \times 3! \times 2!}
\]

### Step 4: Calculate the factorials.
- \(11! = 39916800\)
- \(2! = 2\)
- \(3! = 6\)

Now, substitute these values into the formula:

\[
\text{Permutations} = \frac{39916800}{2! \times 3! \times 2!} = \frac{39916800}{2 \times 6 \times 2} = \frac{39916800}{24} = 1663200
\]

### Conclusion:
The number of permutations of the letters in the word **"phillipines"** is **1,663,200**.

Would you like more details or have any other questions?

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Here are 5 related questions for further exploration:

1. How do you calculate permutations for a word with no repeated letters?
2. What would be the number of permutations for a word with even more repeated letters?
3. How do permutations differ from combinations in mathematics?
4. Can you explain how factorials work and why they are important in permutation calculations?
5. How would the answer change if we used a word with 12 letters instead of 11?

**Tip:** When calculating large factorials, using tools like calculators or software for accuracy is helpful!

Determine the number of permutations of the letters in the word: phillipines

Explore how to calculate the number of permutations of the letters in the word 'phillipines', considering repeated letters like 'i' and 'l'. This combinatorics problem uses factorials and the multiset permutation formula. Suitable for high school students in Grades 9-12.

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