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Question

In a hockey league, every team plays every other team exactly once. For n teams, the number

of hockey games that must be scheduled is given by the expression n2

_____ 2 n

2 ​. Ray’s hockey league

schedules 36 games. How many teams are in Ray’s league?

Answer 1

The combination of n things chosen r at a time is defined as: C(n,r) = n!/(r!(n-r)!)

In this case, we know that we must chose r = 2 teams at a time for each game. We also know that the total number of games played will be C(n,2) = 36.

We can therefore write: C(n,2) = n!/(2!(n-2)!) = 36.

We can multiply both sides by 2! = 2 to get: n!/(n-2)! = 72.

We know that n! = n*(n-1)*(n-2)*(n-3)....

We also know that (n-2)! = (n-2)*(n-3)*(n-4)...

When we divide these terms, everything cancels out except for n*(n-1).

So we are left with n*(n-1) = 72.

If we expand this, we can write, n^2-n-72 = 0.

After factoring, we get: (n-9)(n+8) = 0

This equation is true when n = 9 or when n = -8. However, we cannot have -8 teams! This is an extraneous solution. Therefore, we know our answer must be 9 teams.