Let's address each part of the problem one at a time.

---

(i) We do not want the same flavor more than once:

If we cannot repeat any flavor, the problem becomes choosing 12 distinct flavors out of 31 available. Since the order of the cones doesn't matter, it's a combination problem.

The number of ways is given by: $\binom{31}{12} = \frac{31!}{12! \cdot (31-12)!}$

---

(ii) A flavor may be ordered as many as 12 times:

If there are no restrictions on how many times a flavor can be ordered, it means we can distribute 12 ice cream cones into 31 available flavors with repetition allowed. This is a classic "stars and bars" problem, where the formula is:

$\binom{n+k-1}{k}$ Here:

• $n = 31$ (number of flavors)
• $k = 12$ (number of cones)

So the total number of ways is: $\binom{31+12-1}{12} = \binom{42}{12}$

---

(iii) A flavor may be ordered no more than 11 times:

This is a variation of the "stars and bars" problem with restrictions. The key is that no flavor can be selected more than 11 times.

To solve:

1. First, calculate the unrestricted number of ways (as in part ii): $\binom{42}{12}$.
2. Subtract the cases where one flavor is chosen 12 times, leaving no cones for the other 30 flavors:
   • If one flavor is chosen 12 times, the remaining 0 cones must be distributed among the 30 flavors. There is only 1 way to do this.

Thus: $\text{Total Ways} = \binom{42}{12} - 31$

---

Final Formulas:

1. (i) $\binom{31}{12}$
2. (ii) $\binom{42}{12}$
3. (iii) $\binom{42}{12} - 31$

Let me know if you'd like the numerical evaluations for these!

---

Questions:

1. Would you like detailed calculations of factorials for any of the parts?
2. Are you familiar with the concept of "stars and bars" in combinatorics?
3. Do you want the numerical solutions for these formulas?
4. Would you like a breakdown of the restrictions in part (iii)?
5. Are there any variations of this problem you'd like to explore?

Tip:

When solving combinatorics problems, always clearly identify if the order matters (permutations) or not (combinations) to use the right formula!