Calculate the number of one-to-one functions there are from a set with 6 elements to sets with the following numbers of elements: (a) 5 (b) 6 (c) 7 (d) 8

(a) Since all the elements from the set of 6 have to be transformed to different elements in the resulting set (one-to-one function definition), then you can't have a one-to-one function from the set of 6 elements to the set of 5 elements. Answer: 0.

(b) First element can be transformed into the one of 6, second - one of the 5 that are left, third - one of 4, ..., fifth - one of 2, sixth - to one, so we $6\cdot5\cdot4\cdot3\cdot2\cdot1=6!=720$ functions. Answer: 720.

(c) Same as in b but starting with 7, $7\cdot6\cdot5\cdot4\cdot3\cdot2=5040$ functions. Answer: 5040.

(d) Same as in b but starting with 8, $8\cdot7\cdot6\cdot5\cdot4\cdot3=20160$ functions. Answer: 20160.