Question

How many six-digit numbers are there that are odd and have at least one zero?

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Answer to a math question How many six-digit numbers are there that are odd and have at least one zero?

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Nash

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Riješenje: Broj 6-znamenkastih brojeva koji su neparni i imaju barem jednu nulu jednak je broju 6-znamenkastih brojeva koji su neparni minus broj 6-znamenkastih brojeva koji su neparni i nemaju nulu. Pronalaženje broja 6-znamenkastih brojeva koji su neparni, prva znamenka mora biti neparan broj. Stoga su jedine opcije za prvu znamenku 1, 3, 5, 7 i 9. U međuvremenu, za ostalih 5 znamenki, sve mogu biti bilo što između 0 do 9. Prema tome, broj 6-znamenkastih brojeva koji su neparan je jednako n\lijevo(neparno\desno)=5\cdot10\cdot10\cdot10\cdot10\cdot10 n\lijevo(neparno\desno)=500000 U međuvremenu, za broj šesteroznamenkastih brojeva koji su neparni i nemaju nulu, prva znamenka i dalje mora biti neparan broj. Međutim, za preostalih 5 znamenki, sve mogu biti samo između 1 i 9, jer nijedna znamenka ne može biti nula. Stoga je broj 6-znamenkastih brojeva koji su neparni i nemaju nulu jednak n\lijevo(neparno\:i\:ne\:nula\desno)=5\cdot9\cdot9\cdot9\cdot9\cdot9 n\lijevo(neparno\:i\:ne\:nula\desno)=295245 Stoga je broj 6-znamenkastih brojeva koji su neparni i imaju barem jednu nulu jednak n\lijevo(neparno\:i\:najmanje\:jedan\:nula\desno)=n\lijevo(neparno\desno)-n\lijevo(neparno\:i\:ne\:nula\ desno)

n\lijevo(neparno\:i\:najmanje\:jedan\:nula\desno)=500000-295245

n\lijevo(neparno\:i\:najmanje\:jedan\:nula\desno)=204755 Odgovor: 204755

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