Is selecting the batting order in a baseball game a permutation or combination?

When we say "batting order", we are indicating that we care about which player bats first, then second, etc. and so it's a permutation.

Let's think about this in a slightly different way. Let's say there is a local Little League team that has exactly 9 players. How many ways can we determine the batting order using a permutation and then using a combination.

Permutation

We have 9 players and 9 spots in the batting order.

• In the first spot we can put any of the 9 players.
• In the second spot we can put any of the remaining 8 players.
• In the third spot we can put any of the remaining 7 players.
• And so on.

We can write this as #9xx8xx7xx6xx5xx4xx3xx2xx1=9!#. We can also write this as a permutation, with the general formula being:

#P_(n,k)=(n!)/((n-k)!); n="population", k="picks"#

And so we'd get:

#P_(9,9)=(9!)/((9-9)!)=(9!)/1=9!#

And in the end, whichever way you write it, you get 362,880 ways to set up the batting order.

Combination

With a combination, we don't care in what order the players are placed - we only care that they are in the batting order somewhere. (It's like with a poker hand - we don't care that we drew the Ace of hearts first, the Ace of spades second, the Ace of diamonds third, and the Ace of clubs fourth - all that matters is that we have 4-of-a-kind in Aces).

So let's look at our Little League club again. There are 9 players and 9 spots and so each player will be in the lineup. And so there is only 1 combination possible.

We can see it in the equation - the general equation of which is:

#C_(n,k)=(n!)/((k!)(n-k)!)# with #n="population", k="picks"#

and so we get:

#C_(9,9)=(9!)/((9!)(9-9)!)=(9!)/((9!)(0!))=1#