We must move to a given position in the given number of moves; each move is either to the left or right. How many distinct paths are there?
So, to move over one spot in 9 moves requires 5 right (RRRRR) and 4 left (LLLL). The question simplifies down to the following:
How many different strings can be made by rearranging RRRRRLLLL?
This simplifies even further to this: How many different ways can we assign 5 R’s to 9 spots? Since once the R’s are given, the L’s positions are known — the places remaining. And we can go from the other way: How many ways can we assign 4 L’s to 9 spots? It’s the same thing, again because assigning L’s is assigning R’s.
The solution is found using the binomial coefficient function .
The factorial form is .