How many one-to-one functions are there from a set with two elements to a set with four elements?

12

Step-by-step explanation:

There are 4*3 = 12 ways that two of the four elements can be mapped to the two elements of the smaller set. Here's a list of the 12 possible functions:

{ (a, 1), (b, 2}}, { (a, 2), (b, 1) }, { (a, 1), (b, 3) }, { (a, 3), (b, 1) },

{ (a, 1), (b, 4) }, { (a, 4), (b, 1) }, { (a, 2), (b, 3) }, { (a, 3), (b, 2) },

{ (a, 2), (b, 4) }, { (a, 4}, (b, 2) }, { (a, 3), (b, 4) }, { (a, 4), (b, 3) }