"In the Champions League Round of 8 draw on some year, the clubs present are: Barcelona, Bayern, Benfica, Inter Milan, Liverpool, Man City, PSG and Real Madrid.
a) Of all the draws possible, in which ones we have Barcelona and Bayern facing each other?
b) Of all the draws possible, in which ones we have Inter Milan facing an English club?"
I know it looks simple, but i'm having some troubles.
edit: I know that the number of possible draws are $$\dfrac{8!}{2^{4}\times 4!} = \frac{8\times7\times6\times5\times4!}{2^{4}\times4!} = 7\times3\times5=105$$
Then, given that the draws that one determinated club will face another club are equal (there are no restrictions whatsoever), then the draws that pits barcelona and bayern against each other is 15?
It's $\dfrac{1}{7}$ of the possible draws all along? :O
for letter b) i'm still drawing a blank.
edit2: I just found this is the formula of the double factorial.
You solved the first problem correctly.
Suppose the teams are listed alphabetically. Barcelona would then be the first team on the list. There are seven ways to select an opponent for Barcelona. That leaves six teams. There are five ways to select an opponent for the first team remaining in the list. That leaves four teams. There are three ways to select an opponent for the first team remaining in the list. The remaining two teams must play each other. Therefore, there are $$7!! = 7 \cdot 5 \cdot 3 \cdot 1 = 105$$ possible draws for the round of $8$.
If Barcelona plays Bayern, we have to arrange matches for the remaining six teams, which can be done in $$5!! = 5 \cdot 3 \cdot = 15$$ ways, as you found.
If Inter Milan plays one of the two English clubs, there are two ways to select its opponent. Once that opponent is selected, we have to arrange matches among the remaining six teams, which can be done in $5!!$ ways. Hence, there are $$2 \cdot 5!! = 2 \cdot 5 \cdot 3 \cdot 1 = 30$$ possible draws for the round of $8$ in which Inter Milan plays an English club.
If you think in terms of symmetry, notice that $1/7$th of the possible draws have Barcelona playing Bayern in the round of $8$. Similarly, $2/7$ of the possible draws have Inter Milan playing an English club in the round of $8$.