How many permutations exist of the word "PASCALS"? Leave your answer in factorial form.

1. Break Down the Problem

To find the number of permutations of the letters in the word "PASCALS", we need to consider the total number of letters and how many times each letter occurs.

2. Relevant Concepts

The formula for finding the number of permutations of a multiset is given by:

$\frac{n!}{n_1! \cdot n_2! \cdot n_k!}$

Where:

• $n$ is the total number of letters.
• $n_1, n_2, \ldots, n_k$ are the frequencies of the distinct letters.

3. Analysis and Detail

The word "PASCALS" consists of 8 letters. However, the letter 'A' appears twice. The frequencies of the letters are as follows:

• P: 1
• A: 2
• S: 2
• C: 1
• L: 1
• T: 1

Thus, we have:

• Total letters $n = 8$
• Frequencies: $n_A = 2, n_S = 2, n_P = 1, n_C = 1, n_L = 1$

4. Verify and Summarize

Using the formula for permutations of multisets:

$\text{Permutations} = \frac{8!}{2! \cdot 2! \cdot 1! \cdot 1! \cdot 1!}$

Final Answer

The number of permutations of the word "PASCALS" in factorial form is:

$\frac{8!}{2! \cdot 2!}$