"A football team consists of 20 offensive and 20 defensive players. The players are to be paired in groups of 2 for the purpose of determining roommates. If the pairing is done at random what is the probability that there are no offensive-defensive pairings?"
My first thoughts were that there are (40Cr2)=720 possible pairings although i'm told its much larger than this...
Any ideas? Any prods in the right direction would be much appreciated.
using patterns i would imagine the amount of OOs and DDs is
(20!)/((2^10))(10!)) and that the probability would be this divided by your number.
However could you explain how you arrived at this expression?
(p.s. is there a tutorial anywhere on how to use the Math Input function on this forum... the thing denoted by the capital Sigma)
Here is a very quick and dirty way to explain this.
Say we have a class of 20 freshmen.
If we divide them up into four study groups: biology, mathematics, literature, and history.
That can be done in ways. ( N choose k.)
Those are known as ordered partitions. Because it makes a difference to a student which group he/she is in.
Think a class of 20 freshmen is calculus.
If we divide them up into four groups to review a just returned test.
Those are known as unordered partitions because only content matters.
This can be done ways.
The details are a bit much to into.
But I hope you can begin to get the idea.