Show that there is a language A subset $\{0,1{\}}^{*}$ with the following properties:

$(1)$ For all $xinA,|x|5.$

$(2)$ No DFA with fewer than $9$ states decides $A.$

Count the number of languages satisfying $(1)$ and the number of DFAs with fewer than $9$ states, and argue that there aren't enough DFAs to decide all those languages. The former should be greater than the latter. Show your work, and state a concrete number in each case. To count the number of languages satisfying $(1),$ think about writing down all the strings of length at most $5,$ and then to define such a language, you have to make a binary decision for each string about whether to include it in the language or not. How many ways are there to make these choices?

To count the number of DFAs with at most $8$ states, consider that a DFA behaves identically even if you rename all the states, so you can assume without loss of generality that any DFA with k states has the state set $\{$q$1,$ q$2,...,$qk$\}.$ Now think about how to count how many ways there are to choose the other parts of the DFA. It's okay to overcount here: we might identify two DFAs as different even though they decide the same language. For instance, adding extra unreachable states to a DFA does not alter the language it decides.$)$ This overcounting is okay as long as this number is smaller than the number of languages satisfying $(1),$ since the number of distinct languages decided by the DFAs we count will be even smaller than the number of DFAs we count.