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Question Number 124237 by mr W last updated on 01/Dec/20

How many 6 digit odd numbers have  different digits?

$${How}\:{many}\:\mathrm{6}\:{digit}\:{odd}\:{numbers}\:{have} \\ $$$${different}\:{digits}? \\ $$

Commented by mr W last updated on 02/Dec/20

5×(P_5 ^9 −P_4 ^8 )=67200

$$\mathrm{5}×\left({P}_{\mathrm{5}} ^{\mathrm{9}} −{P}_{\mathrm{4}} ^{\mathrm{8}} \right)=\mathrm{67200} \\ $$

Answered by liberty last updated on 01/Dec/20

case(1) A□□□□B where A and B odd number   = 5×P _4^8  ×4   case(2) A□□□□B where A even and B odd number  = 4× P _4^8 ×5   totally = P _4^8  × 40 = ((8×7×6×5×4!)/(4!)) ×40   = 56×30×40=67200  56×30×40  67200.0

$${case}\left(\mathrm{1}\right)\:{A}\Box\Box\Box\Box{B}\:{where}\:{A}\:{and}\:{B}\:{odd}\:{number} \\ $$$$\:=\:\mathrm{5}×{P}\:_{\mathrm{4}} ^{\mathrm{8}} \:×\mathrm{4}\: \\ $$$${case}\left(\mathrm{2}\right)\:{A}\Box\Box\Box\Box{B}\:{where}\:{A}\:{even}\:{and}\:{B}\:{odd}\:{number} \\ $$$$=\:\mathrm{4}×\:{P}\:_{\mathrm{4}} ^{\mathrm{8}} ×\mathrm{5}\: \\ $$$${totally}\:=\:{P}\:_{\mathrm{4}} ^{\mathrm{8}} \:×\:\mathrm{40}\:=\:\frac{\mathrm{8}×\mathrm{7}×\mathrm{6}×\mathrm{5}×\mathrm{4}!}{\mathrm{4}!}\:×\mathrm{40} \\ $$$$\:=\:\mathrm{56}×\mathrm{30}×\mathrm{40}=\mathrm{67200} \\ $$$$\mathrm{56}×\mathrm{30}×\mathrm{40} \\ $$$$\mathrm{67200}.\mathrm{0} \\ $$$$ \\ $$

Commented by mr W last updated on 02/Dec/20

correct sir, thanks!

$${correct}\:{sir},\:{thanks}! \\ $$

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