Question-55468

Question Number 55468 by peter frank last updated on 24/Feb/19

Answered by mr W last updated on 25/Feb/19

to choose 4 characters there are three cases: case 1: two letters, one digit, one symbol to choose 2 letters there are C_2 ^(26) =325 ways, to choose a digit there are 9 ways, to choose a symbol there are 5 ways, totally 325×9×5=14625 ways case 2: one letter, two digits, one symbol to choose 1 letter there are 26 way, to choose 2 digits there are C_2 ^9 =36 ways, to choose a symbol there are 5 ways, totally 26×36×5=4680 ways case 3: one letter, one digit, two symbols to choose 1 letter there are 26 way, to choose 1 digit there are 9 ways, to choose 2 symbols there are C_2 ^5 =10 ways, totally 26×9×10=2340 ways ⇒ to choose 4 characters there are 14625+4680+2340=21645 ways. to arrange each 4 characters there are 4! ways, therefore total number of passwords which can be formed is 21645×4!=519480

$${to}\:{choose}\:\mathrm{4}\:{characters}\:{there}\:{are}\:{three} \\ $$$${cases}: \\ $$$${case}\:\mathrm{1}:\:{two}\:{letters},\:{one}\:{digit},\:{one}\:{symbol} \\ $$$${to}\:{choose}\:\mathrm{2}\:{letters}\:{there}\:{are}\:{C}_{\mathrm{2}} ^{\mathrm{26}} =\mathrm{325}\:{ways}, \\ $$$${to}\:{choose}\:{a}\:{digit}\:{there}\:{are}\:\mathrm{9}\:{ways}, \\ $$$${to}\:{choose}\:{a}\:{symbol}\:{there}\:{are}\:\mathrm{5}\:{ways}, \\ $$$${totally}\:\mathrm{325}×\mathrm{9}×\mathrm{5}=\mathrm{14625}\:{ways} \\ $$$$ \\ $$$${case}\:\mathrm{2}:\:{one}\:{letter},\:{two}\:{digits},\:{one}\:{symbol} \\ $$$${to}\:{choose}\:\mathrm{1}\:{letter}\:{there}\:{are}\:\mathrm{26}\:{way}, \\ $$$${to}\:{choose}\:\mathrm{2}\:{digits}\:{there}\:{are}\:{C}_{\mathrm{2}} ^{\mathrm{9}} =\mathrm{36}\:{ways}, \\ $$$${to}\:{choose}\:{a}\:{symbol}\:{there}\:{are}\:\mathrm{5}\:{ways}, \\ $$$${totally}\:\mathrm{26}×\mathrm{36}×\mathrm{5}=\mathrm{4680}\:{ways} \\ $$$$ \\ $$$${case}\:\mathrm{3}:\:{one}\:{letter},\:{one}\:{digit},\:{two}\:{symbols} \\ $$$${to}\:{choose}\:\mathrm{1}\:{letter}\:{there}\:{are}\:\mathrm{26}\:{way}, \\ $$$${to}\:{choose}\:\mathrm{1}\:{digit}\:{there}\:{are}\:\mathrm{9}\:{ways}, \\ $$$${to}\:{choose}\:\mathrm{2}\:{symbols}\:{there}\:{are}\:{C}_{\mathrm{2}} ^{\mathrm{5}} =\mathrm{10}\:{ways}, \\ $$$${totally}\:\mathrm{26}×\mathrm{9}×\mathrm{10}=\mathrm{2340}\:{ways} \\ $$$$ \\ $$$$\Rightarrow\:{to}\:{choose}\:\mathrm{4}\:{characters}\:{there}\:{are} \\ $$$$\mathrm{14625}+\mathrm{4680}+\mathrm{2340}=\mathrm{21645}\:{ways}. \\ $$$$ \\ $$$${to}\:{arrange}\:{each}\:\mathrm{4}\:{characters}\:{there} \\ $$$${are}\:\mathrm{4}!\:{ways},\:{therefore}\:{total}\:{number} \\ $$$${of}\:{passwords}\:{which}\:{can}\:{be}\:{formed}\:{is} \\ $$$$\mathrm{21645}×\mathrm{4}!=\mathrm{519480} \\ $$

Commented by peter frank last updated on 25/Feb/19