You have 12 different types of candies and you want to choose a selection that contains at least one candy. In how many different ways can you choose one or more candies from these 12 distinct types?

Introduction / Context:
This problem involves counting subsets of a set of distinct items, in this case 12 different types of candies. We are asked to count all selections that contain at least one candy, meaning empty selections are not allowed. This is a standard application of the relationship between the total number of subsets and the number of non empty subsets.

Given Data / Assumptions:

• There are 12 distinct types of candies.
• Each type can be either chosen or not chosen in a selection.
• We want selections that include at least one type of candy.
• We do not limit the maximum number of types; we can choose anywhere from 1 up to all 12 types.
• Candies are considered by type; choosing the same type multiple times is not part of this counting.

Concept / Approach:
For a set of n distinct elements, the total number of subsets (including the empty set) is 2^n. This counts every pattern of choosing or not choosing each element. To find the number of non empty subsets, we subtract the single empty subset from the total. Therefore, for n items, the number of non empty subsets is 2^n - 1. Here, n = 12.

Step-by-Step Solution:
Step 1: Compute the total number of subsets of a 12 element set: 2^12. Step 2: 2^12 = 4096. Step 3: Among these subsets, exactly one is the empty set, which contains no candies. Step 4: The problem asks for selections with at least one candy, so we must exclude the empty set. Step 5: Therefore, number of non empty subsets = 2^12 - 1. Step 6: 4096 - 1 = 4095. Step 7: Hence, there are 4095 different ways to choose one or more candies.

Verification / Alternative check:
We can also think of each candy type as a binary choice: either included or not included in the selection. Each of the 12 types contributes a factor of 2 choices. Multiplying gives 2 * 2 * ... * 2 (12 times) = 2^12 = 4096 total patterns. Only one pattern has all types excluded, so removing that pattern leaves 4095 valid non empty selections. This matches the earlier reasoning exactly.

Why Other Options Are Wrong:
4054, 4050, 4059: These numbers are all less than 4095 and do not have any special relation to powers of 2 for n = 12. None of them equal 2^12 - 1, which is the correct formula.

Common Pitfalls:
A common mistake is to forget the empty set and answer 4096 instead of 4095. Others may try to sum combinations for choosing 1, 2, 3, ..., 12 types separately, which is correct but time consuming. If done correctly, that sum should also equal 2^12 - 1. Remembering the subset formula and the simple subtraction of the empty set provides a fast and reliable method.

Final Answer:
The number of different ways to choose one or more types of candy from 12 distinct types is 4095.