Let A 1 2 8 and B 1 2 2000 Find the number of fun

. Let A = {1, 2, . . . , 8} and B = {1, 2, . . . , 2000}. Find the number of functions that there are from A to B that have image that either is a one-element set or is a set that does not contain the element 1.

Solution

First we count functions whose image is a one-element set

So such functions are ones where all elements of A are mapped to one element of B.

So there are 2000 functions of B for each element

Now we count functions which do not contain 1

So we ignore 1 and count function from A to B-{1}

1 can be mapped to one of 1999 elements

2 can also be mapped to one of 1999 elements

and so on.

so, total: 1999^8 functions are possible.