# Combinatronics Problem...

• Apr 10th 2009, 10:33 AM
sebjory
Combinatronics Problem...
"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.
• Apr 10th 2009, 11:41 AM
Plato
Quote:

Originally Posted by sebjory
"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?"

The are $\frac{40!}{(2^{20})(20!)}$ random pairings.
Can you finish?
• Apr 10th 2009, 01:10 PM
sebjory
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?

Many thanks

(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)
• Apr 10th 2009, 01:56 PM
Plato
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 $\binom{20}{5}\binom{15}{5} \binom{10}{5} \binom{5}{5} =\frac{20!}{(5!)^4}$ ways. ( $\binom{N}{k}$ 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 $\frac{20!}{(5!)^4(4!)}$ ways.
The details are a bit much to into.
But I hope you can begin to get the idea.

Learn LaTeX