1.
Consider a group of n objects. Assume we want to form groups of r, from these n objects.
There are in total [tex]C(n, r)= \frac{n!}{r!(n-r)!} [/tex] many ways of doing so.
where, r! is "r factorial", calculated as 1*2*3*...*(r-1)*r
2.
C(x, 2) is the total number of pairs out of x objects, that we can form.
so let the x objects represent the x teams, and 2, represent a group of 2, which means a game.
The total number of games is:
n(x)=2*C(x, 2) = [tex]2* \frac{x!}{2!(x-2)!}= \frac{x!}{(x-2)!}= \frac{x(x-1)(x-2)!}{(x-2)!}=x(x-1)= x^{2} -x [/tex]
Remark, we multiplied by 2, since there will be 2 matches for each pair of teams.
Answer: [tex] x^{2} -x[/tex]
