a) How many social security cards can be issued with no repeated digits? b) How many social security cards that can be issued with at least one digit repeated? Exercise 4. a) How many nonempty subsets are in a set of 5 elements? b) Erika has 5 friends. In how many ways can she invite one or more friends to a dinner party?

Answer:a) How many social security cards can be issued with no repeated digits?We have 10 numbers from 0 to 9. When no digit is repeated, so we get 10! ways that is = [tex]10\times9\times8\times7\times6\times5\times4\times3\times2\times1[/tex]= 3628800b) How many social security cards that can be issued with at least one digit repeated? We will get this by subtracting 10! from all possible cases.[tex]10^{9}-10![/tex]= [tex]1000000000-3628800=996371200[/tex] Exercise 4. a) How many nonempty subsets are in a set of 5 elements? Each element has 2 choices that are either selected in subset or not selected in subset.Here, total choices will be [tex]2^{5}[/tex]And it is 1 time that no element is selected.So, the number of non empty subsets will be = [tex]2^{5}-1[/tex]=> [tex]32-1=31[/tex]b) Erika has 5 friends. In how many ways can she invite one or more friends to a dinner party?Same like above part, here Erika has two choices for each friend, either invite or not invite.So, she has total [tex]2^{5}=32[/tex] choices.1 way to not invite anyone.Hence, the number of ways can she invite one or more friends to a dinner party = [tex]32-1=31[/tex] ways.