SOLUTION: Please can you help me solve this work problem question. How many one to one functions are there from set with m elements to one with n elements?

Let me start with this DEFINITION


    A function f is 1 -to- 1 if no two elements in the domain of f correspond to the same element in the range of f . 
    In other words, each x in the domain has exactly one image in the range; and, no y in the range is the image 
    of more than one x in the domain.



Therefore, if the function is one-to-one from the set M of "m" elements to the set N of "n" elements, then NECESSARY  n >= m.



Now, for the 1-st element in M, we can choose its image in N by n different ways;

     for the 2-nd element in M, we cam choose its image in N by (n-1) different ways among the remaining (n-1) elements in N;

     for the 3-rd element in M, we cam choose its image in N by (n-2) different ways among the remaining (n-2) elements in N;


     . . . . and so on, till the last element in M.



Therefore, the number of all one-to-one functions from M to N is

     n*(n-1)*(n-2)* . . . * (n-m+1).

(the product of m integer factors in descending order, starting from n).     ANSWER