Let r, l be two integers such that r l 3 . What is the total number of functions f : 1, 2, , r 1, 2, , r such that f(1), f(2), , f(l) are all distinct?

Step-by-step solution

The total number of elements in the domain is r and in the codomain is r . We need to find the number of functions such that the images of the first l elements, f(1), f(2), , f(l) , are all distinct. The number of ways to choose distinct images for these l elements from the r available elements in the codomain is given by ^ r P_ l : ^ r P_ l = r(r-1)(r-2) (r-l+1) For the remaining r-l elements in the domain, there are no restrictions. Each of these elements can be mapped to any of the r elements in the codomain. Th