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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

$a s Q 55468, b u t p a s s w o r d s w i t h 6$ $c h a r a c t e r s .$
	Answered by ajfour last updated on 25/Feb/19

	$1, 228, 500, 000 .$ $m u s t b e c o r r e c t S i r .$ $4, 1, 1 \Rightarrow^{26} C_{4} \times 9 \times 5 = 672750$ $1, 4, 1 \Rightarrow 26 \times^{9} C_{4} \times 5 = 16380$ $1, 1, 4 \Rightarrow 26 \times 9 \times^{5} C_{4} = 1170$ $3, 2, 1 \Rightarrow^{26} C_{3} \times^{9} C_{2} \times 5 = 468000$ $1, 2, 3 \Rightarrow 26 \times^{9} C_{2} \times^{5} C_{3} = 9360$ $2, 3, 1 \Rightarrow^{26} C_{2} \times^{9} C_{3} \times 5 = 136500$ $1, 3, 2 \Rightarrow 26 \times^{5} C_{3} \times^{5} C_{2} = 21840$ $2, 1, 3 \Rightarrow^{26} C_{2} \times 9 \times^{5} C_{3} = 29250$ $3, 1, 2 \Rightarrow^{26} C_{3} \times 9 \times^{5} C_{2} = 234000$ $2, 2, 2 \Rightarrow^{26} C_{2} \times^{9} C_{2} \times^{5} C_{2} = 117000$ $- - - - - -$ $1706250$ $- - - - - -$ $1706250 \times 6! = 1, 228, 500, 000$ $_________________________.$
	Commented by mr W last updated on 25/Feb/19

	$a b s o l u t e l y c o r r e c t s i r! t h a n k s a l o t!$
	Answered by mr W last updated on 26/Feb/19

	$a l t e r n a t i v e w a y f o r s o l u t i o n :$ $f o r l o n g p a s s w o r d s w i t h m o r e a n d m o r e$ $c h a r a c t e r s, t h e m e t h o d w i t h h e l p o f$ $g e n e r a t i n g f u n c t i o n i s t h e b e t t e r w a y$ $t o s o l v e t h e q u e s t i o n .$ $6 c h a r a c t e r s c a n b e c h o o s e n l i k e t h i s :$ $o n e o r m o r e f r o m 26 l e t t e r s$ $o n e o r m o r e f r o m 9 d i g i t s$ $o n e o r m o r e f r o m 5 s y m b o l s$ $i f w e c h o o s e k f r o m 26 l e t t e r s, w e h a v e$ $C_{k}^{26} w a y s t o d o t h a t . t h e c o r r e s p o n d i n g$ $g e n e r a t i n g f u n c t i o n i s$ $C_{1}^{26} x + C_{2}^{26} x^{2} + C_{3}^{26} x^{3} + . . . + C_{26}^{26} x^{26}$ $= C_{0}^{26} x^{0} + C_{1}^{26} x + C_{2}^{26} x^{2} + C_{3}^{26} x^{3} + . . . + C_{26}^{26} x^{26} - 1$ $= {(1 + x)}^{26} - 1$ $s i m i l a r l y t h e g e n e r a t i n g f u n c t i o n$ $f o r c h o o s i n g d i g i t s i s$ $C_{1}^{9} x + C_{2}^{9} x^{2} + C_{3}^{9} x^{3} + . . . + C_{9}^{9} x^{9}$ $= {(1 + x)}^{9} - 1$ $t h e g e n e r a t i n g f u n c t i o n f o r s y m b o l s i s$ $C_{1}^{5} x + C_{2}^{5} x^{2} + C_{3}^{5} x^{3} + . . . + C_{5}^{5} x^{5}$ $= {(1 + x)}^{5} - 1$ $t h e t o t a l g e n e r a t i n g f u n c t i o n i s$ $[{(1 + x)}^{26} - 1] [{(1 + x)}^{9} -] [{(1 + x)}^{5} - 1]$ $t h e c o e f f i c i e n t o f t e r m x^{6} i s t h e$ $n u m b e r o f w a y s t o c h o o s e 6 c h a r a c t e r s .$ $a f t e r e x p e n d i n g t h e f u n c t i o n w e g e t$
	Commented by mr W last updated on 25/Feb/19

	Commented by mr W last updated on 26/Feb/19

	$t h e c o e f f i c i e n t o f t e r m x^{6} i s 1706250,$ $i . e . t h e r e a r e 1706250 w a y s t o s e l e c t$ $6 c h a r a c t e r s . s i n c e t h e r e a r e 6! w a y s$ $t o a r r a n g e 6 c h a r a c t e r s i n d i f f e r e n t$ $o r d e r, n u m b e r o f p o s s i b l e p a s s w o r d s i s$ $1706250 \times 6! = 1 228 500 000 .$ $s i m i l a r l y t h e n u m b e r o f p a s s w o r d s$ $w i t h 3 c h a r a c t e r s i s$ $1170 \times 3! = 7020 .$ $t h e n u m b e r o f p a s s w o r d s$ $w i t h 4 c h a r a c t e r s i s$ $21645 \times 4! = 519480 .$ $t h e n u m b e r o f p a s s w o r d s$ $w i t h 10 c h a r a c t e r s i s$ $625039701 \times 10! \approx 2.268 \times 10^{15} .$ $t h e n u m b e r o f p a s s w o r d s$ $w i t h 20 c h a r a c t e r s i s$ $134514143575 \times 20! \approx 3.273 \times 10^{28} .$
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