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Question Number 43343 by pieroo last updated on 10/Sep/18

how many odd numbers greater than 60000 can be made from the digits 5,6,7,8,9,0 if no number contains any digit more than once?

$$\mathrm{how}\:\mathrm{many}\:\mathrm{odd}\:\mathrm{numbers}\:\mathrm{greater}\:\mathrm{than}\:\mathrm{60000}\:\mathrm{can}\:\mathrm{be}\:\mathrm{made} \\ $$$$\mathrm{from}\:\mathrm{the}\:\mathrm{digits}\:\mathrm{5},\mathrm{6},\mathrm{7},\mathrm{8},\mathrm{9},\mathrm{0}\:\mathrm{if}\:\mathrm{no}\:\mathrm{number}\:\mathrm{contains} \\ $$$$\mathrm{any}\:\mathrm{digit}\:\mathrm{more}\:\mathrm{than}\:\mathrm{once}? \\ $$

Answered by tanmay.chaudhury50@gmail.com last updated on 10/Sep/18

−_5 −_4 −_3 −_2 −_1 digits are 5,6,7,8,9,0 (six nos of digit) numbers greater than 60,000... the 5th place can be filled by (6,7,8,9) numbers are odd so 1st place can be filled(5,7,9) 1)−_5 ^6 −_4 −_3 −_2 −_1 ^5 5th place filled by 6 and 1st place by5 4th place can be filled by any digit out of(7,8,9,0) so 4ways 3rd place can be filled by any remaining three digit so 3ways 2nd place can be filled by any remaining two digit so 2 ways Thus the numbers whose 1st place is 5 and 5th place is 6 that means greater than 60000 but odd are 4×3×2=24 −_5 ^6 −_4 ^× −_3 ^× −_2 ^× −_1 ^5 →24 −_5 ^6 −_4 ^× −_3 ^× −_2 ^× −_1 ^7 →24 −_5 ^6 −_4 ^× −_3 ^× −_2 ^× −_1 ^9 −24 thus 24×3=72 −_5 ^7 −_4 ^× −_3 ^× −_2 ^× −_1 ^5 →24 −_5 ^7 −_4 ^× −_3 ^× −_2 ^× −_1 ^9 →24 −_5 ^8 ^ −_4 ^× −_3 ^× −_2 ^× −_1 ^5 →24 −_5 ^8 −_4 ^× −_3 ^× −_2 ^× −_1 ^7 →24 −_5 ^8 −_4 ^× −_3 ^× −_2 ^× −_1 ^9 →24 −_5 ^9 −_4 ^× −_3 ^× −_2 ^× −_1 ^5 →24 −_5 ^9 −_4 ^× −_3 ^× −_2 ^× −_1 ^7 →24 5th place 6 →24×3=72 5th placd 7→24×2=48 5th place 8→24×3=72 5th place 9→24×2=48 total=72+48+72+48=240

$$\underset{\mathrm{5}} {−}\:\underset{\mathrm{4}} {−}\:\underset{\mathrm{3}} {−}\:\underset{\mathrm{2}} {−}\:\underset{\mathrm{1}} {−}\: \\ $$$${digits}\:{are}\:\mathrm{5},\mathrm{6},\mathrm{7},\mathrm{8},\mathrm{9},\mathrm{0}\:\:\left({six}\:{nos}\:{of}\:{digit}\right) \\ $$$${numbers}\:{greater}\:{than}\:\mathrm{60},\mathrm{000}... \\ $$$${the}\:\mathrm{5}{th}\:{place}\:{can}\:{be}\:{filled}\:{by}\:\left(\mathrm{6},\mathrm{7},\mathrm{8},\mathrm{9}\right) \\ $$$${numbers}\:{are}\:{odd}\:{so}\:\mathrm{1}{st}\:{place}\:{can}\:{be}\:{filled}\left(\mathrm{5},\mathrm{7},\mathrm{9}\right) \\ $$$$\left.\mathrm{1}\right)\underset{\mathrm{5}} {\overset{\mathrm{6}} {−}}\:\underset{\mathrm{4}} {−}\:\underset{\mathrm{3}} {−}\:\underset{\mathrm{2}} {−}\:\underset{\mathrm{1}} {\overset{\mathrm{5}} {−}}\:\mathrm{5}{th}\:{place}\:{filled}\:{by}\:\mathrm{6}\:{and}\:\mathrm{1}{st}\:{place}\:{by}\mathrm{5} \\ $$$$\mathrm{4}{th}\:{place}\:{can}\:{be}\:{filled}\:{by}\:{any}\:{digit}\:{out}\:{of}\left(\mathrm{7},\mathrm{8},\mathrm{9},\mathrm{0}\right) \\ $$$${so}\:\mathrm{4}{ways} \\ $$$$\mathrm{3}{rd}\:{place}\:{can}\:{be}\:{filled}\:{by}\:{any}\:{remaining}\:{three}\:{digit} \\ $$$${so}\:\mathrm{3}{ways} \\ $$$$\mathrm{2}{nd}\:{place}\:{can}\:{be}\:{filled}\:{by}\:{any}\:{remaining}\:{two} \\ $$$${digit}\:{so}\:\mathrm{2}\:{ways} \\ $$$$\mathcal{T}{hus}\:{the}\:{numbers}\:{whose}\:\mathrm{1}{st}\:{place}\:{is}\:\mathrm{5} \\ $$$${and}\:\mathrm{5}{th}\:{place}\:{is}\:\mathrm{6}\:\:{that}\:{means}\:{greater}\:{than} \\ $$$$\mathrm{60000}\:{but}\:{odd}\:{are}\:\mathrm{4}×\mathrm{3}×\mathrm{2}=\mathrm{24} \\ $$$$\underset{\mathrm{5}} {\overset{\mathrm{6}} {−}}\:\:\underset{\mathrm{4}} {\overset{×} {−}}\:\underset{\mathrm{3}} {\overset{×} {−}}\:\underset{\mathrm{2}} {\overset{×} {−}}\:\underset{\mathrm{1}} {\overset{\mathrm{5}} {−}}\:\rightarrow\mathrm{24} \\ $$$$\underset{\mathrm{5}} {\overset{\mathrm{6}} {−}}\:\underset{\mathrm{4}} {\overset{×} {−}}\:\underset{\mathrm{3}} {\overset{×} {−}}\:\underset{\mathrm{2}} {\overset{×} {−}}\:\underset{\mathrm{1}} {\overset{\mathrm{7}} {−}}\rightarrow\mathrm{24} \\ $$$$\underset{\mathrm{5}} {\overset{\mathrm{6}} {−}}\:\underset{\mathrm{4}} {\overset{×} {−}}\:\underset{\mathrm{3}} {\overset{×} {−}}\:\underset{\mathrm{2}} {\overset{×} {−}}\:\underset{\mathrm{1}} {\overset{\mathrm{9}} {−}}−\mathrm{24} \\ $$$${thus}\:\mathrm{24}×\mathrm{3}=\mathrm{72} \\ $$$$\underset{\mathrm{5}} {\overset{\mathrm{7}} {−}}\:\underset{\mathrm{4}} {\overset{×} {−}}\:\underset{\mathrm{3}} {\overset{×} {−}}\:\underset{\mathrm{2}} {\overset{×} {−}}\:\underset{\mathrm{1}} {\overset{\mathrm{5}} {−}}\:\rightarrow\mathrm{24} \\ $$$$\underset{\mathrm{5}} {\overset{\mathrm{7}} {−}}\:\underset{\mathrm{4}} {\overset{×} {−}}\:\underset{\mathrm{3}} {\overset{×} {−}}\:\underset{\mathrm{2}} {\overset{×} {−}}\:\underset{\mathrm{1}} {\overset{\mathrm{9}} {−}}\:\rightarrow\mathrm{24} \\ $$$$\underset{\mathrm{5}} {\overset{\mathrm{8}} {−}}\overset{} {\:}\underset{\mathrm{4}} {\overset{×} {−}}\:\underset{\mathrm{3}} {\overset{×} {−}}\:\underset{\mathrm{2}} {\overset{×} {−}}\:\underset{\mathrm{1}} {\overset{\mathrm{5}} {−}}\rightarrow\mathrm{24} \\ $$$$\underset{\mathrm{5}} {\overset{\mathrm{8}} {−}}\:\underset{\mathrm{4}} {\overset{×} {−}}\:\underset{\mathrm{3}} {\overset{×} {−}}\:\underset{\mathrm{2}} {\overset{×} {−}}\:\underset{\mathrm{1}} {\overset{\mathrm{7}} {−}}\rightarrow\mathrm{24} \\ $$$$\:\underset{\mathrm{5}} {\overset{\mathrm{8}} {−}}\:\underset{\mathrm{4}} {\overset{×} {−}}\:\underset{\mathrm{3}} {\overset{×} {−}}\:\underset{\mathrm{2}} {\overset{×} {−}}\:\underset{\mathrm{1}} {\overset{\mathrm{9}} {−}}\rightarrow\mathrm{24} \\ $$$$\underset{\mathrm{5}} {\overset{\mathrm{9}} {−}}\:\underset{\mathrm{4}} {\overset{×} {−}}\:\underset{\mathrm{3}} {\overset{×} {−}}\:\underset{\mathrm{2}} {\overset{×} {−}}\:\underset{\mathrm{1}} {\overset{\mathrm{5}} {−}}\rightarrow\mathrm{24} \\ $$$$\underset{\mathrm{5}} {\overset{\mathrm{9}} {−}}\:\underset{\mathrm{4}} {\overset{×} {−}}\underset{\mathrm{3}} {\overset{×} {−}}\:\underset{\mathrm{2}} {\overset{×} {−}}\:\underset{\mathrm{1}} {\overset{\mathrm{7}} {−}}\rightarrow\mathrm{24} \\ $$$$\mathrm{5}{th}\:{place}\:\mathrm{6}\:\rightarrow\mathrm{24}×\mathrm{3}=\mathrm{72} \\ $$$$\mathrm{5}{th}\:{placd}\:\mathrm{7}\rightarrow\mathrm{24}×\mathrm{2}=\mathrm{48} \\ $$$$\mathrm{5}{th}\:{place}\:\mathrm{8}\rightarrow\mathrm{24}×\mathrm{3}=\mathrm{72} \\ $$$$\mathrm{5}{th}\:{place}\:\mathrm{9}\rightarrow\mathrm{24}×\mathrm{2}=\mathrm{48} \\ $$$${total}=\mathrm{72}+\mathrm{48}+\mathrm{72}+\mathrm{48}=\mathrm{240} \\ $$$$ \\ $$$$ \\ $$$$\:\:\:\:\: \\ $$

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