How many distinct permutations can be formed using the letters of the word 

​"ENTERTAINENTERTAIN​"?

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There are 

enter your response here 

distinct permutations.

To find the number of distinct permutations of the letters in the word "ENTERTAINENTERTAIN," we need to account for the repetitions of each letter.

First, we count the total number of letters and the frequency of each distinct letter.

The word "ENTERTAINENTERTAIN" has 18 letters. The frequencies of each letter are as follows:
- E: 4 times
- N: 4 times
- T: 4 times
- R: 2 times
- A: 2 times
- I: 2 times

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

\[
\frac{n!}{n_1! \cdot n_2! \cdot \ldots \cdot n_k!}
\]

where \( n \) is the total number of letters, and \( n_1, n_2, \ldots, n_k \) are the frequencies of the repeating letters.

For the word "ENTERTAINENTERTAIN":
- Total number of letters, \( n = 18 \)
- Frequencies: E = 4, N = 4, T = 4, R = 2, A = 2, I = 2

So, the number of distinct arrangements is:

\[
\frac{18!}{4! \cdot 4! \cdot 4! \cdot 2! \cdot 2! \cdot 2!}
\]

Calculating the factorials:

\[
18! = 6402373705728000
\]
\[
4! = 24
\]
\[
2! = 2
\]

Substituting these values into the formula:

\[
\frac{18!}{4! \cdot 4! \cdot 4! \cdot 2! \cdot 2! \cdot 2!} = \frac{6402373705728000}{24 \cdot 24 \cdot 24 \cdot 2 \cdot 2 \cdot 2}
\]

Simplifying the denominator:

\[
24 \cdot 24 \cdot 24 = 13824
\]
\[
2 \cdot 2 \cdot 2 = 8
\]
\[
13824 \cdot 8 = 110592
\]

Finally:

\[
\frac{6402373705728000}{110592} = 57827515584000
\]

Therefore, there are \( 57827515584000 \) distinct permutations.

Do you want any further details or have any questions?

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Tip: When dealing with large factorials, using a calculator or software can help avoid errors and make the computation more manageable.

Learn how to calculate the number of distinct permutations of the letters in the word 'ENTERTAINENTERTAIN'. This detailed guide covers the concept of permutations with repeating elements and provides a step-by-step solution. Suitable for high school students.

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