Let's break this down step by step:

Part 1: Order is considered, no flavor repeated

If the order in which the flavors are chosen matters and no flavor can be repeated, we are dealing with permutations. The formula for the number of permutations of $r$ items from a set of $n$ distinct items is:

$P(n, r) = \frac{n!}{(n - r)!}$

In this case, we are choosing 6 flavors from 29 available flavors, so we apply the formula with $n = 29$ and $r = 6$:

$P(29, 6) = \frac{29!}{(29 - 6)!} = \frac{29!}{23!}$

Now, calculating the factorial:

$P(29, 6) = 29 \times 28 \times 27 \times 26 \times 25 \times 24 = 11022480$

So, the number of possible 6-dip sundaes when order matters and no flavor is repeated is 11,022,480.

Part 2: Order is not considered, no flavor repeated

If order doesn't matter and no flavor can be repeated, we are dealing with combinations. The formula for the number of combinations of $r$ items from a set of $n$ distinct items is:

$C(n, r) = \frac{n!}{r!(n - r)!}$

In this case, we are choosing 6 flavors from 29, so we apply the formula with $n = 29$ and $r = 6$:

$C(29, 6) = \frac{29!}{6!(29 - 6)!} = \frac{29!}{6!23!}$

We can now calculate the result:

$C(29, 6) = \frac{29 \times 28 \times 27 \times 26 \times 25 \times 24}{6 \times 5 \times 4 \times 3 \times 2 \times 1}$

Let's compute this:

$C(29, 6) = \frac{29 \times 28 \times 27 \times 26 \times 25 \times 24}{720} = 475020$

So, the number of possible 6-dip sundaes when order does not matter and no flavor is repeated is 475,020.

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Summary:

• Part 1: 11,022,480 sundaes (order matters).
• Part 2: 475,020 sundaes (order doesn't matter).

Would you like to explore further or need any clarification?

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Here are some related questions for practice:

1. How many 5-dip sundaes are possible if order is considered, and no flavor is repeated?
2. How many 4-dip sundaes are possible if order is not considered, and no flavor is repeated?
3. If 30 ice cream flavors were available, how would the number of possible sundaes change?
4. If repetition of flavors were allowed, how would the total number of sundaes differ?
5. How would the formula change if more than one sundae was being selected at once?

Tip: Always check if the problem asks for permutations (order matters) or combinations (order does not matter) to decide which formula to apply.