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

Answer: 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