How many? one-to-one correspondences are there between two sets with 7 elements? each?

5040

This is the same as asking "how many ways are there to order 7 items?" which is equal to 7! = 7*6*5*4*3*2*1 = 5040

Or you can use the nPr formula to get
nPr = (n!)/(n-r)!
7P7 = (7!)/(7-7)!
7P7 = (7!)/(0!)
7P7 = (7*6*5*4*3*2*1)/(1)
7P7 = 5040/1
7P7 = 5040
leading to the same answer