Description
Question
Refer back to the Learning Activities titled A Formula for Permutations and A Formula for Combinations. Explain how the Fundamental Counting Principle is used each time the Permutation Formula (nPr) or the Combination Formula (nCr) is applied. Would these be considered independent or dependent events? Or, is it inappropriate to be concerned about whether it is independent/dependent? Explain your thinking on the question
My Response
The fundamental counting principle states that suppose one of the events has m likely outcomes and the second has n likely outcomes, then possible outcomes totals to m*n events when combined. Besides, a combination denotes the number of ways of selecting k objects from the n total number of objects (the order does not matter). In this instance, the fundamental principle is applied in permutation or combination formula when counting the number of ways we can perform an activity without counting manually. To event A and B on their independence, the calculation would involve P(A), P(B), and P(A ? B), and then confirm whether P(A ? B) equals P(A)P(B). If they are equivalent, A and B are independent. However, if they are not, they are dependent. In this instance, the selection of Toy A, then B, C, and D is the same combination as selecting Toy C first, then B, D, and A. For this reason, they are independent events.
Feedback
I’m going to say that I think your first part response is fine. Where you say “the fundamental principle is applied in permutation or combination formula when counting the number of ways we can perform an activity”, I would point to the n! in the denominator, which is doing exactly that. I’d like to focus the rest of the response a little bit:
Q1: Do permutations involve independent or dependent events?
Q2: Do combinations involve independent or dependent events?
There will be one additional question after this, but it makes more sense if we wait to hear your response to these questions.
My response to the Feedback
Permutations and combinations are the two likely possibilities of combining or arranging the outcomes of independent events. The groupings in which the order of the items matters are called permutations. On the other hand, combinations are groupings in which the content matters, but the order of the items does not. From what we know about dependent events situations, one action eliminates possible future actions. Another important aspect to consider about dependent events outcomes is how the events are organized. According to the fundamental counting principle, the number of outcomes in a sample space is equated to the product of the number of outcomes for each event. For this reason, permutations involve dependent events. On the other hand, computations involve independent events.
Feedback
Hi, Ahmad. Again, thank you for the response. Both permutation and combinations have an underlying condition that you cannot replace what you select. As a result they both involve dependent events. In baseball jargon, you went one for two. 🙂
Now, that brings up an interesting question: “How is it possible to use the Fundamental Counting Principle (which requires independent events) within something like the Permutation Formula (which requires dependent events)?” Let’s use as example 5 P 3 = (5)(4)(3) = 60. That’s not an algebraic simplification of the Permutation Formula. It’s actually a conceptual way to do that calculation using the Fundamental Counting Principle. But then they would have to be independent events! The ‘5’ is not the problem; the first selection is always an independent event. How can we describe the ‘4’ and the ‘3’ in such a way that they are also independent events?
