Question

There are two sets S and T. |S|=4 and |T|=10, how many one to one functions and onto functions can be made from these sets?

Answer 1

For one to one functions, each element of S will be mapped to only one distinct element of T. That is we need to select 4 elements out of 10 elements of T where ordering of selection is important.

Number of one to one functions = ¹⁰P₄ = 10! / (10-4)! = 10! / 6!

= 10 * 9 * 8 * 7

= 5040

For onto functions, each element of S is mapped to at least one element of T.

Total number of ways in which 4 elements of T can be uniquely mapped to 4 elements of S = ⁴P₄  = 24

Total number of ways in which rest 6 elements can be mapped to any of 4 elements of S = 4⁶

Total number of onto functions = 24 * 4⁶ = 98304