Counting Functions

Each element in set can be mapped to any element in set . The set has 4 elements, thus the total number of functions is .

Now, we need to exclude functions where , meaning none of 's elements map to 1. For this, the remaining options are 4, 9, or 16. Therefore, each of the 4 elements in has 3 choices, leading to such functions.

Thus, the functions where are 81. Consequently, the number of functions where is .

However, we need many-one functions, meaning at least two elements in map to the same element in . A function that is one-to-one does not satisfy this, and such functions amount to the number of unique permutations of mappings from to with 4 elements having 4 and each mapped uniquely, i.e., .

Thus, the many-one functions where is .

Hence, the number of many-one functions where is 151.