Functions Part 1

There are 16 ordered pairs in S×S. We write all these ordered pairs in 4 sets as follows.
A={(1,1)}
B={(1,4),(2,4),(3,4)(4,4),(4,3),(4,2),(4,1)}
C={(1,3),(2,3),(3,3),(3,2),(3,1)}
D={(1,2),(2,2),(2,1)}
All elements of set B have image 4 and only element of A has image 1 .
All elements of set C have image 3 or 4 and all elements of set D have image 2 or 3 or 4 .
We will solve this question in two cases.
Case I: When no element of set C has image 3 .
Number of onto functions =2 (when elements of set D have images 2 or 3 )
Case II: When atleast one element of set C has image 3 .
Number of onto functions =(23−1)(1+2+2) =35
Total number of functions =37