What does a k out of n alternatives constraint require?

The k out of n alternatives constraint establishes a condition in decision-making problems where a specific number of selections (k) must be made from a larger set of options (n). In this context, the correct assertion is that the sum of the n binary variables must equal k.

In practical terms, binary variables represent the inclusion (1) or exclusion (0) of each alternative. By ensuring that the sum of these binary variables equals k, the constraint guarantees that exactly k alternatives are chosen from the n total available options. This is crucial for problems where decision-makers need to select a defined number of options but cannot exceed or fall short of that number.

The other options are not consistent with the logical requirements of a k out of n constraint. The first option incorrectly suggests that the sum of k binary variables must equal n, which does not fit the scenario of selecting k alternatives from n available choices. The third option implies each variable must exceed k, and the last option limits the selection to a single alternative, which contradicts the essence of having multiple choices. Thus, the requirement for the sum of the n binary variables to equal k accurately captures the intention behind the k out of n constraint.