Combinatorics Question

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Combinatorics Question

Mathematics
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At vero eos et accusamus et iusto odio dignissimos ducimus qui blanditiis praesentium voluptatum deleniti atque corrupti quos dolores et quas molestias excepturi sint occaecati cupiditate non provident, similique sunt in culpa qui officia deserunt mollitia animi, id est laborum et dolorum fuga. Et harum quidem rerum facilis est et expedita distinctio. Nam libero tempore, cum soluta nobis est eligendi optio cumque nihil impedit quo minus id quod maxime placeat facere possimus, omnis voluptas assumenda est, omnis dolor repellendus. Itaque earum rerum hic tenetur a sapiente delectus, ut aut reiciendis voluptatibus maiores alias consequatur aut perferendis doloribus asperiores repellat.

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Eight people enter a competition. In how many ways can `a first prize`, `a second prize`, `and three identical third prizes` be awarded to these eight people?
This was a quiz from a previous week. The teacher has since given us the answer, but I think she made a typo. I wanted to see if someone could verify.
So one way to do it, this ways makes more sense to me... Choose the first and second place people, so this would be 2-permutations of 8 people. \(\large\rm P(8,2)=\frac{8!}{6!}\) And for the remaining three places, since they're identical, order will not matter, and we're simply looking at combinations of these uhh remaining slots. \(\rm \left(\begin{matrix}6\\3\end{matrix}\right)=\dfrac{6!}{3!3!}\) Answer = 1120.

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The other way the teacher showed it... was to do \(\large\rm \left(\begin{matrix}8\\5\end{matrix}\right)P(5,2)=\dfrac{8!}{5!3!}\cdot\dfrac{5!}{3!}\) Hmm this one also equals out to 1120. I can't seem to make sense of it though. So we're ... choose 5 people from 8, not regarding order at this point. And then we're doing 2-permutations of the 5 we selected to place in first and second? Is that making sense?
Ya maybe I didn't need to open up a question for this one -_- shoulda just thought it through lol I think it's making more sense now
lol' who are you talking to Mr.ghost?

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