To find the number of ways to choose an ice cream cone with three scoops of ice cream, all different flavors, from six available flavors, where order doesn't matter, you can use combinations. The formula for combinations is:

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

where:

• $n$ is the total number of items (in this case, flavors),
• $r$ is the number of items to choose (in this case, scoops of ice cream),
• $!$ represents the factorial of a number.

Here, $n = 6$ and $r = 3$.

$\binom{6}{3} = \frac{6!}{3!(6 - 3)!} = \frac{6 \times 5 \times 4}{3 \times 2 \times 1} = \frac{120}{6} = 20$

Thus, there are 20 ways to choose 3 different flavors from 6, where the order does not matter.

Would you like a detailed explanation or have any questions?

Here are 5 related questions:

1. How would the calculation change if two flavors could be the same?
2. How many ways can you choose three flavors if the order does matter?
3. If the parlor introduces two more flavors, how many ways can you choose 3 different flavors?
4. How many ways can you choose 4 different flavors from the same 6 flavors?
5. How many combinations are possible if you are allowed to pick only 2 flavors?

Tip: When order doesn't matter, use combinations; when order matters, use permutations!