Question

Combinatorics Question

Answer 1

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?

Answer 2

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.

Answer 3

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.

Answer 4

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?

Answer 5

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

Answer 6

lol'
who are you talking to Mr.ghost?