Deciding first, second, third baseman and shortstop out of 6 infielders
has 6x5x4=120 ways.
Ordering nine starting batters have 9x8x7...x3x2x1=9! ways which are
362880 ways.! is referred to a factorial and it is calculated by multiplying
itself and all positive integers smaller than itself.
n!=n·(n− 1 )·(n− 2 ).... 3 · 2 · 1
Let's look at another way of selecting players.
There are 10 ways of deciding 3 outfielders out of 5 outfielders
5 × 34 !× (^3) = 10
There are 5 ways of deciding 4 infielders out of 6 infielders
4!
6 × 5 × (^4) = 5
Have you discovered the difference between two calculations? The first
method has decided players in a specific position, LF, CF, and RF, which
means that it considers an order of the positions. This is called
permutation.
Permutation of choosing r amount out of n is the following. (r≤n)
(^) nPr=n(n− 1 )(n− 2 )···×(n−r+ 1 )
(^) nPr=(nn−!r)!
The second method was using a combination. We have not separated
first, second, third baseman and shortstop with their positions, keeping
as one position of ‘infielder’, and selecting players without considering
the order.
Combination of choosing r amount out of n is the following. (r≤n)
(^) nCr=^ nrP!r