call that hexadecimal numbers are constructed using the $16$ digits $0,1,2,3,4,5,6,7,8,9,$ A$,$ B$,$ C$,$ D$,$ E$,$ F$.$

$($a$)$

How many strings of hexadecimal digits consist of from one through four digits?

The set of hexadecimal digits consisting of from one through four digits can be thought of as broken up into disjoint subsets:

S$1,$

S$2,$

and so forth, where each

Sk

consists of the strings with k hexadecimal digits. There are

strings in

S$1,$

strings in

S$2,$

and so forth. Thus, by the $---$Select$---$addition$$multiplication rule, the total number of strings is