Baseball Batting Orders

Date: 10/14/2001 at 18:27:28
From: Erica 
Subject: Word problem! 

How many possible batting orders are there for a team of nine players? 
 
I have no clue how to even start figuring this out. Please help!

Date: 10/15/2001 at 15:19:12
From: Doctor Ian
Subject: Re: Word problem! 

Hi Erica,

Often when a problem seems too big to tackle, you can make some 
progress by considering a smaller problem.  

Suppose there is only 1 player (a). Then there is only 1 possible 
order, right? 

  a

What if we add a second player (b)? He could go either before or after 
a, right? 

  a -> ba
       ab

So there are two possible orders for two players. What if we add a 
third player (c)? He could go in any position in any of the orders we 
already had:

  a -> ba -> cba
             bca
             bac
       ab -> cab
             acb
             abc

So now we have 1*2*3 different orders. Let's try it for one more 
player (d):

  a -> ba -> cba -> dcba
                    cdba
                    cbda
                    cbad
             bca -> dbca
                    bdca
                    bcda
                    bcad
             bac -> [4 more orders]
       ab -> cab -> [4 more orders]
             acb -> [4 more orders]
             abc -> [4 more orders]

Now we have 1*2*3*4 different orders. 

It looks as if when we add the n'th player, we multiply the number of 
possible orders by n (because there are n possible changes to each of 
the orders we had before). So if we keep going until we have 9 
players, how many possible orders would we have?

Does this help? 

- Doctor Ian, The Math Forum
  http://mathforum.org/dr.math/