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Question Number 55502 by mr W last updated on 25/Feb/19
Commented by mr W last updated on 25/Feb/19
$${as}\:{Q}\mathrm{55468},\:{but}\:{passwords}\:{with}\:\mathrm{6} \\ $$$${characters}. \\ $$
Answered by ajfour last updated on 25/Feb/19
$$\mathrm{1},\mathrm{228},\mathrm{500},\mathrm{000}\:. \\ $$$${must}\:{be}\:{correct}\:{Sir}. \\ $$$$\mathrm{4},\mathrm{1},\mathrm{1}\Rightarrow\:^{\mathrm{26}} {C}_{\mathrm{4}} ×\mathrm{9}×\mathrm{5}=\mathrm{672750} \\ $$$$\mathrm{1},\mathrm{4},\mathrm{1}\Rightarrow\:\mathrm{26}×^{\mathrm{9}} {C}_{\mathrm{4}} ×\mathrm{5}=\:\mathrm{16380} \\ $$$$\mathrm{1},\mathrm{1},\mathrm{4}\:\Rightarrow\:\mathrm{26}×\mathrm{9}×^{\mathrm{5}} {C}_{\mathrm{4}} =\:\:\mathrm{1170} \\ $$$$\mathrm{3},\mathrm{2},\mathrm{1}\:\Rightarrow\:^{\mathrm{26}} {C}_{\mathrm{3}} ×^{\mathrm{9}} {C}_{\mathrm{2}} ×\mathrm{5}=\mathrm{468000} \\ $$$$\mathrm{1},\mathrm{2},\mathrm{3}\:\Rightarrow\:\mathrm{26}×^{\mathrm{9}} {C}_{\mathrm{2}} ×^{\mathrm{5}} {C}_{\mathrm{3}} =\mathrm{9360} \\ $$$$\mathrm{2},\mathrm{3},\mathrm{1}\:\Rightarrow\:^{\mathrm{26}} {C}_{\mathrm{2}} ×^{\mathrm{9}} {C}_{\mathrm{3}} ×\mathrm{5}=\mathrm{136500} \\ $$$$\mathrm{1},\mathrm{3},\mathrm{2}\:\Rightarrow\:\mathrm{26}×^{\mathrm{5}} {C}_{\mathrm{3}} ×^{\mathrm{5}} {C}_{\mathrm{2}} =\mathrm{21840} \\ $$$$\mathrm{2},\mathrm{1},\mathrm{3}\:\Rightarrow\:^{\mathrm{26}} {C}_{\mathrm{2}} ×\mathrm{9}×^{\mathrm{5}} {C}_{\mathrm{3}} =\mathrm{29250} \\ $$$$\mathrm{3},\mathrm{1},\mathrm{2}\:\Rightarrow\:^{\mathrm{26}} {C}_{\mathrm{3}} ×\mathrm{9}×^{\mathrm{5}} {C}_{\mathrm{2}} =\mathrm{234000} \\ $$$$\mathrm{2},\mathrm{2},\mathrm{2}\:\Rightarrow\:^{\mathrm{26}} {C}_{\mathrm{2}} ×^{\mathrm{9}} {C}_{\mathrm{2}} ×^{\mathrm{5}} {C}_{\mathrm{2}} =\mathrm{117000} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:−−−−−− \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\mathrm{1706250} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:−−−−−− \\ $$$$\:\:\:\mathrm{1706250}×\mathrm{6}!\:=\mathrm{1},\mathrm{228},\mathrm{500},\mathrm{000} \\ $$$$\:\:\:\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_. \\ $$
Commented by mr W last updated on 25/Feb/19
$${absolutely}\:{correct}\:{sir}!\:{thanks}\:{alot}! \\ $$
Answered by mr W last updated on 26/Feb/19
![alternative way for solution: for long passwords with more and more characters, the method with help of generating function is the better way to solve the question. 6 characters can be choosen like this: one or more from 26 letters one or more from 9 digits one or more from 5 symbols if we choose k from 26 letters, we have C_k ^(26) ways to do that. the corresponding generating function is C_1 ^(26) x+C_2 ^(26) x^2 +C_3 ^(26) x^3 +...+C_(26) ^(26) x^(26) =C_0 ^(26) x^0 +C_1 ^(26) x+C_2 ^(26) x^2 +C_3 ^(26) x^3 +...+C_(26) ^(26) x^(26) −1 =(1+x)^(26) −1 similarly the generating function for choosing digits is C_1 ^9 x+C_2 ^9 x^2 +C_3 ^9 x^3 +...+C_9 ^9 x^9 =(1+x)^9 −1 the generating function for symbols is C_1 ^5 x+C_2 ^5 x^2 +C_3 ^5 x^3 +...+C_5 ^5 x^5 =(1+x)^5 −1 the total generating function is [(1+x)^(26) −1][(1+x)^9 −][(1+x)^5 −1] the coefficient of term x^6 is the number of ways to choose 6 characters. after expending the function we get](Q55517.png)
$${alternative}\:{way}\:{for}\:{solution}: \\ $$$${for}\:{long}\:{passwords}\:{with}\:{more}\:{and}\:{more} \\ $$$${characters},\:{the}\:{method}\:{with}\:{help}\:{of} \\ $$$${generating}\:{function}\:{is}\:{the}\:{better}\:{way} \\ $$$${to}\:{solve}\:{the}\:{question}. \\ $$$$\mathrm{6}\:{characters}\:{can}\:{be}\:{choosen}\:{like}\:{this}: \\ $$$${one}\:{or}\:{more}\:{from}\:\mathrm{26}\:{letters} \\ $$$${one}\:{or}\:{more}\:{from}\:\mathrm{9}\:{digits} \\ $$$${one}\:{or}\:{more}\:{from}\:\mathrm{5}\:{symbols} \\ $$$${if}\:{we}\:{choose}\:{k}\:{from}\:\mathrm{26}\:{letters},\:{we}\:{have} \\ $$$${C}_{{k}} ^{\mathrm{26}} \:{ways}\:{to}\:{do}\:{that}.\:{the}\:{corresponding} \\ $$$${generating}\:{function}\:{is} \\ $$$${C}_{\mathrm{1}} ^{\mathrm{26}} {x}+{C}_{\mathrm{2}} ^{\mathrm{26}} {x}^{\mathrm{2}} +{C}_{\mathrm{3}} ^{\mathrm{26}} {x}^{\mathrm{3}} +...+{C}_{\mathrm{26}} ^{\mathrm{26}} {x}^{\mathrm{26}} \\ $$$$={C}_{\mathrm{0}} ^{\mathrm{26}} {x}^{\mathrm{0}} +{C}_{\mathrm{1}} ^{\mathrm{26}} {x}+{C}_{\mathrm{2}} ^{\mathrm{26}} {x}^{\mathrm{2}} +{C}_{\mathrm{3}} ^{\mathrm{26}} {x}^{\mathrm{3}} +...+{C}_{\mathrm{26}} ^{\mathrm{26}} {x}^{\mathrm{26}} −\mathrm{1} \\ $$$$=\left(\mathrm{1}+{x}\right)^{\mathrm{26}} −\mathrm{1} \\ $$$${similarly}\:{the}\:{generating}\:{function} \\ $$$${for}\:{choosing}\:{digits}\:{is} \\ $$$${C}_{\mathrm{1}} ^{\mathrm{9}} {x}+{C}_{\mathrm{2}} ^{\mathrm{9}} {x}^{\mathrm{2}} +{C}_{\mathrm{3}} ^{\mathrm{9}} {x}^{\mathrm{3}} +...+{C}_{\mathrm{9}} ^{\mathrm{9}} {x}^{\mathrm{9}} \\ $$$$=\left(\mathrm{1}+{x}\right)^{\mathrm{9}} −\mathrm{1} \\ $$$${the}\:{generating}\:{function}\:{for}\:{symbols}\:{is} \\ $$$${C}_{\mathrm{1}} ^{\mathrm{5}} {x}+{C}_{\mathrm{2}} ^{\mathrm{5}} {x}^{\mathrm{2}} +{C}_{\mathrm{3}} ^{\mathrm{5}} {x}^{\mathrm{3}} +...+{C}_{\mathrm{5}} ^{\mathrm{5}} {x}^{\mathrm{5}} \\ $$$$=\left(\mathrm{1}+{x}\right)^{\mathrm{5}} −\mathrm{1} \\ $$$${the}\:{total}\:{generating}\:{function}\:{is} \\ $$$$\left[\left(\mathrm{1}+{x}\right)^{\mathrm{26}} −\mathrm{1}\right]\left[\left(\mathrm{1}+{x}\right)^{\mathrm{9}} −\right]\left[\left(\mathrm{1}+{x}\right)^{\mathrm{5}} −\mathrm{1}\right] \\ $$$${the}\:{coefficient}\:{of}\:{term}\:{x}^{\mathrm{6}} \:{is}\:{the}\: \\ $$$${number}\:{of}\:{ways}\:{to}\:{choose}\:\mathrm{6}\:{characters}. \\ $$$$ \\ $$$${after}\:{expending}\:{the}\:{function}\:{we}\:{get} \\ $$
Commented by mr W last updated on 25/Feb/19
Commented by mr W last updated on 26/Feb/19
$${the}\:{coefficient}\:{of}\:{term}\:{x}^{\mathrm{6}} \:{is}\:\mathrm{1706250}, \\ $$$${i}.{e}.\:{there}\:{are}\:\mathrm{1706250}\:{ways}\:{to}\:{select} \\ $$$$\mathrm{6}\:{characters}.\:{since}\:{there}\:{are}\:\mathrm{6}!\:{ways} \\ $$$${to}\:{arrange}\:\mathrm{6}\:{characters}\:{in}\:{different} \\ $$$${order},\:{number}\:{of}\:{possible}\:{passwords}\:{is} \\ $$$$\mathrm{1706250}×\mathrm{6}!=\mathrm{1}\:\mathrm{228}\:\mathrm{500}\:\mathrm{000}. \\ $$$$ \\ $$$${similarly}\:{the}\:{number}\:{of}\:{passwords} \\ $$$${with}\:\mathrm{3}\:{characters}\:{is} \\ $$$$\mathrm{1170}×\mathrm{3}!=\mathrm{7020}. \\ $$$$ \\ $$$${the}\:{number}\:{of}\:{passwords} \\ $$$${with}\:\mathrm{4}\:{characters}\:{is} \\ $$$$\mathrm{21645}×\mathrm{4}!=\mathrm{519480}. \\ $$$$ \\ $$$${the}\:{number}\:{of}\:{passwords} \\ $$$${with}\:\mathrm{10}\:{characters}\:{is} \\ $$$$\mathrm{625039701}×\mathrm{10}!\approx\mathrm{2}.\mathrm{268}×\mathrm{10}^{\mathrm{15}} . \\ $$$$ \\ $$$${the}\:{number}\:{of}\:{passwords} \\ $$$${with}\:\mathrm{20}\:{characters}\:{is} \\ $$$$\mathrm{134514143575}×\mathrm{20}!\:\approx\mathrm{3}.\mathrm{273}×\mathrm{10}^{\mathrm{28}} . \\ $$