Which statement best describes an independent choice in counting problems?

Which statement best describes an independent choice in counting problems?

• A. Dependent choices where order matters.
• B. Permutations where order matters.
• C. Random sampling without regard to order.
• D. Combinations involving the arrangement of elements into groups without regard to order.

Answer: (D)

In counting problems, independence here means you care only about which elements are chosen, not the order in which they are chosen. When the order doesn’t matter, different sequences that pick the same elements are considered the same outcome. That’s exactly what combinations capture: choosing a set of elements without regard to arrangement. Think of forming a committee by selecting a certain number of people from a group. Whether you pick Alice then Bob or Bob then Alice, you end up with the same committee. Since the order of selection doesn’t create a different result, this is described by combinations—groups where order is irrelevant. The other ideas mix in order or treat the process as something like sampling where order or sequence changes the outcome, which isn’t what independent choices aimed to describe in this context.

In counting problems, independence here means you care only about which elements are chosen, not the order in which they are chosen. When the order doesn’t matter, different sequences that pick the same elements are considered the same outcome. That’s exactly what combinations capture: choosing a set of elements without regard to arrangement.

Think of forming a committee by selecting a certain number of people from a group. Whether you pick Alice then Bob or Bob then Alice, you end up with the same committee. Since the order of selection doesn’t create a different result, this is described by combinations—groups where order is irrelevant.

The other ideas mix in order or treat the process as something like sampling where order or sequence changes the outcome, which isn’t what independent choices aimed to describe in this context.