Here are the detailed solutions to each problem:

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1. Distinguishable Permutations of "ELLIPSES"

The word "ELLIPSES" contains 8 letters:

• E: 2 times
• L: 2 times
• I: 1 time
• P: 1 time
• S: 2 times

The formula for distinguishable permutations is:
$\text{Permutations} = \frac{n!}{p_1! \cdot p_2! \cdot p_3! \cdots}$
where $n!$ is the total number of letters, and $p_1!, p_2!, \ldots$ are the factorials of the frequencies of each repeated letter.

$\text{Permutations} = \frac{8!}{2! \cdot 2! \cdot 2!} = \frac{40320}{8} = 5040$

Answer: 5040 distinguishable permutations.

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2. Number of Ways 6 People Can Sit Around a Circular Table

For a circular arrangement, the formula is:
$\text{Permutations} = (n - 1)!$
where $n$ is the number of people.

$\text{Permutations} = (6 - 1)! = 5! = 120$

Answer: 120 ways.

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3. Permutations of the Digits in "348838"

The number "348838" has 6 digits:

• 3: 2 times
• 8: 3 times
• 4: 1 time

Using the same formula as in Problem 1:
$\text{Permutations} = \frac{n!}{p_1! \cdot p_2! \cdot p_3!}$

$\text{Permutations} = \frac{6!}{2! \cdot 3!} = \frac{720}{12} = 60$

Answer: 60 permutations.

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4. Arrangements of Letters in "MISSISSIPPI"

The word "MISSISSIPPI" contains 11 letters:

• M: 1 time
• I: 4 times
• S: 4 times
• P: 2 times

Using the same formula:
$\text{Permutations} = \frac{n!}{p_1! \cdot p_2! \cdot p_3! \cdot p_4!}$

$\text{Permutations} = \frac{11!}{1! \cdot 4! \cdot 4! \cdot 2!} = \frac{39916800}{1152} = 34650$

Answer: 34650 arrangements.

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5. 4-Digit Numbers from {1, 3, 5, 6, 8, 9} Without Repetition

To form a 4-digit number, we select and arrange 4 digits from a set of 6 digits:

The number of permutations is given by:
$P(n, r) = \frac{n!}{(n-r)!}$

$P(6, 4) = \frac{6!}{(6-4)!} = \frac{720}{2} = 360$

Answer: 360 four-digit numbers.

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Do you want more detailed steps or have any questions?

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5 Related Questions:

1. What is the general formula for circular permutations with identical objects?
2. How does repetition affect the total number of permutations?
3. If a family of 8 sat at a round table, how many arrangements would there be?
4. How many permutations are possible if the digits of "12345" are arranged in groups of 3?
5. What if repetition is allowed in forming 4-digit numbers from {1, 3, 5, 6, 8, 9}? How many combinations are there?

Tip:

For circular permutations, always remember to subtract one from the total number of objects since one position is fixed for rotation symmetry!