In how many ways can the word successful be arranged

To find the number of distinct ways to arrange the letters in the word "SUCCESSFUL," we need to consider the fact that some letters are repeated.

The word "SUCCESSFUL" has 10 letters in total. The frequency of each letter is as follows:

- S: 3 times
- U: 2 times
- C: 2 times
- E: 1 time
- F: 1 time
- L: 1 time

The formula to calculate the number of distinct permutations of a word with repeated letters is:

\[
\text{Number of arrangements} = \frac{n!}{n_1! \times n_2! \times \dots \times n_k!}
\]

Where:
- \(n\) is the total number of letters,
- \(n_1, n_2, \dots, n_k\) are the frequencies of the repeated letters.

For "SUCCESSFUL," this becomes:

\[
\text{Number of arrangements} = \frac{10!}{3! \times 2! \times 2! \times 1! \times 1! \times 1!}
\]

Now, let's calculate it:

\[
10! = 10 \times 9 \times 8 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1 = 3,628,800
\]

\[
3! = 3 \times 2 \times 1 = 6
\]

\[
2! = 2 \times 1 = 2
\]

\[
1! = 1
\]

Thus:

\[
\text{Number of arrangements} = \frac{3,628,800}{6 \times 2 \times 2 \times 1 \times 1 \times 1} = \frac{3,628,800}{24} = 151,200
\]

So, the word "SUCCESSFUL" can be arranged in **151,200** distinct ways.

Would you like further details or have any questions?

Here are some related questions:
1. How would the number of arrangements change if another "S" was added?
2. What if one of the letters was removed? How many arrangements then?
3. How does the factorial function impact the total number of arrangements?
4. What is the significance of dividing by the factorials of repeated letters?
5. Can you calculate the arrangements if no letters were repeated?

**Tip:** When dealing with permutations of repeated elements, always consider the frequency of each repeated item to avoid overcounting.

Learn how to calculate the number of distinct arrangements for the word 'SUCCESSFUL'. Understand the concept of permutations with repeated elements using factorials. Suitable for high school students studying permutations and combinatorics.

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