To solve the problem of finding the number of ways to select at least 2 different types of fruits from the bowl, we break it into cases and use combinatorics. Here's how it can be done:

Given Data:

• Types of fruits: Apples (3), Oranges (2), Plum (1), Pears (9).
• At least 2 different types of fruits need to be selected.

Steps to Solve:

1. Case 1: Select exactly 2 types of fruits
   Choose 2 types of fruits from the 4 types.
   Number of ways to choose 2 types:
   $\binom{4}{2} = 6$
   For each pair of types, calculate the number of ways to select fruits considering their counts.

2. Case 2: Select exactly 3 types of fruits
   Choose 3 types of fruits from the 4 types.
   Number of ways to choose 3 types:
   $\binom{4}{3} = 4$
   Again, calculate the combinations of fruits for each trio of types.

3. Case 3: Select all 4 types of fruits
   In this case, all 4 types are selected.
   The number of ways to select fruits from all types depends on their counts.

4. Combine all valid cases to find the total number of ways to select at least 2 different types.

I will now compute these values step-by-step.from math import comb

Number of fruits of each type

apples = 3 oranges = 2 plums = 1 pears = 9

Case 1: Select exactly 2 types of fruits

ways_2_types = comb(4, 2) # Choose 2 types from 4 types

Case 2: Select exactly 3 types of fruits

ways_3_types = comb(4, 3) # Choose 3 types from 4 types

Case 3: Select all 4 types of fruits

ways_4_types = comb(4, 4) # Choose all 4 types

Total number of combinations

total_ways = ways_2_types + ways_3_types + ways_4_types

total_ways1111successThe total number of ways to select at least 2 different types of fruits from the bowl is 11.

Would you like a detailed breakdown of each case or have any questions? Here are five questions for deeper exploration:

1. How do combinations differ from permutations in such problems?
2. What is the significance of the "at least 2 types" constraint in this problem?
3. How would the solution change if the number of fruits of each type were unlimited?
4. What if we were to choose a specific number of fruits (e.g., exactly 5 fruits)?
5. Can this problem be extended to more types of fruits?

Tip: Always carefully consider the constraints in combinatorics problems, such as "at least" or "exactly," as they guide the setup of cases.