Two separate sets each contain six different items. In how many ways can we select a collection of items such that at least one item is chosen from each set?

Introduction / Context:
This question tests your understanding of counting subsets with an “at least one” condition applied to two different groups. We must count all possible selections that include at least one item from the first set and at least one item from the second set. This is a classic application of subtracting unwanted cases from the total number of subsets.

Given Data / Assumptions:

Set A has 6 distinct items.

Set B has 6 distinct items.

We select any subset of items from the union of A and B.

At least one item must come from A and at least one item from B.

Concept / Approach:
Without any restrictions, each of the 12 items (6 from A, 6 from B) can either be chosen or not chosen. That gives 2^12 possible subsets. We must subtract the subsets that violate the requirement: subsets with no item from A, subsets with no item from B, and then add back the subset with no items from both sets, since it was subtracted twice.

Step-by-Step Solution:
Step 1: Total subsets with no restriction = 2^12 = 4096.Step 2: Subsets with no item from A mean we choose only from B. There are 2^6 = 64 such subsets.Step 3: Subsets with no item from B mean we choose only from A. There are also 2^6 = 64 such subsets.Step 4: The empty set (choosing nothing) has been subtracted twice, once in each of the previous two steps, so we must add it back once.Step 5: Required count = 4096 - 64 - 64 + 1 = 4096 - 128 + 1 = 3969.

Verification / Alternative check:
We can think of independently choosing a nonempty subset from A and a nonempty subset from B and then combining them. The number of nonempty subsets of a 6-element set is 2^6 - 1 = 63. Thus, the total number of valid selections is 63 * 63 = 3969, which confirms our earlier result.

Why Other Options Are Wrong:
2856 does not correspond to any simple combination of powers of 2 here and likely reflects partial subtraction of invalid cases.
480 is far too small and might come from mistakenly using combinations like 6C1 or 6C2 instead of considering all subset sizes.

4096 is the total number of subsets with no restriction and fails to enforce the “at least one from each set” condition.

Common Pitfalls:
Many students forget to add back the empty set when applying the inclusion exclusion principle. Others incorrectly remove subsets that have no items from A or B without thinking about overlaps. A safer approach is often to think in terms of selecting nonempty subsets from each set and multiplying the counts, which correctly handles the condition that both sets contribute at least one element.

Final Answer:
The number of valid selections is 3969.