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

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

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

$$\underset{5}{-}\underset{4}{-}\underset{3}{-}\underset{2}{-}\underset{1}{-}$$$$digitsare5,6,7,8,9,0\left(sixnosofdigit\right)$$$$numbersgreaterthan60,000...$$$$the5thplacecanbefilledby(6,7,8,9)$$$$numbersareoddso1stplacecanbefilled(5,7,9)$$$$1)\underset{5}{\stackrel{6}{-}}\underset{4}{-}\underset{3}{-}\underset{2}{-}\underset{1}{\stackrel{5}{-}}5thplacefilledby6and1stplaceby5$$$$4thplacecanbefilledbyanydigitoutof(7,8,9,0)$$$$so4ways$$$$3rdplacecanbefilledbyanyremainingthreedigit$$$$so3ways$$$$2ndplacecanbefilledbyanyremainingtwo$$$$digitso2ways$$$$\mathcal{T}husthenumberswhose1stplaceis5$$$$and5thplaceis6thatmeansgreaterthan$$$$60000butoddare4\times 3\times 2=24$$$$\underset{5}{\stackrel{6}{-}}\underset{4}{\stackrel{\times }{-}}\underset{3}{\stackrel{\times }{-}}\underset{2}{\stackrel{\times }{-}}\underset{1}{\stackrel{5}{-}}\to 24$$$$\underset{5}{\stackrel{6}{-}}\underset{4}{\stackrel{\times }{-}}\underset{3}{\stackrel{\times }{-}}\underset{2}{\stackrel{\times }{-}}\underset{1}{\stackrel{7}{-}}\to 24$$$$\underset{5}{\stackrel{6}{-}}\underset{4}{\stackrel{\times }{-}}\underset{3}{\stackrel{\times }{-}}\underset{2}{\stackrel{\times }{-}}\underset{1}{\stackrel{9}{-}}-24$$$$thus24\times 3=72$$$$\underset{5}{\stackrel{7}{-}}\underset{4}{\stackrel{\times }{-}}\underset{3}{\stackrel{\times }{-}}\underset{2}{\stackrel{\times }{-}}\underset{1}{\stackrel{5}{-}}\to 24$$$$\underset{5}{\stackrel{7}{-}}\underset{4}{\stackrel{\times }{-}}\underset{3}{\stackrel{\times }{-}}\underset{2}{\stackrel{\times }{-}}\underset{1}{\stackrel{9}{-}}\to 24$$$$\underset{5}{\stackrel{8}{-}}\stackrel{}{}\underset{4}{\stackrel{\times }{-}}\underset{3}{\stackrel{\times }{-}}\underset{2}{\stackrel{\times }{-}}\underset{1}{\stackrel{5}{-}}\to 24$$$$\underset{5}{\stackrel{8}{-}}\underset{4}{\stackrel{\times }{-}}\underset{3}{\stackrel{\times }{-}}\underset{2}{\stackrel{\times }{-}}\underset{1}{\stackrel{7}{-}}\to 24$$$$\underset{5}{\stackrel{8}{-}}\underset{4}{\stackrel{\times }{-}}\underset{3}{\stackrel{\times }{-}}\underset{2}{\stackrel{\times }{-}}\underset{1}{\stackrel{9}{-}}\to 24$$$$\underset{5}{\stackrel{9}{-}}\underset{4}{\stackrel{\times }{-}}\underset{3}{\stackrel{\times }{-}}\underset{2}{\stackrel{\times }{-}}\underset{1}{\stackrel{5}{-}}\to 24$$$$\underset{5}{\stackrel{9}{-}}\underset{4}{\stackrel{\times }{-}}\underset{3}{\stackrel{\times }{-}}\underset{2}{\stackrel{\times }{-}}\underset{1}{\stackrel{7}{-}}\to 24$$$$5thplace6\to 24\times 3=72$$$$5thplacd7\to 24\times 2=48$$$$5thplace8\to 24\times 3=72$$$$5thplace9\to 24\times 2=48$$$$total=72+48+72+48=240$$$$$$$$$$$$$$

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