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Question Number 180421 by mr W last updated on 12/Nov/22
$${find}\:{the}\:{number}\:{of}\:{numbers}\:{less}\:{than}\: \\ $$$$\mathrm{10}^{\mathrm{6}} \:{which}\:{contain}\:{at}\:{least}\:\mathrm{3}\:{different} \\ $$$${digits}. \\ $$
Answered by liuxinnan last updated on 12/Nov/22
$${maybe}\:\mathrm{89965} \\ $$
Commented by liuxinnan last updated on 12/Nov/22
Commented by mr W last updated on 13/Nov/22
$${thanks}\:{for}\:{trying}\:{sir}! \\ $$
Answered by mr W last updated on 13/Nov/22
$${from}\:\mathrm{100}\:{to}\:\mathrm{999999}\:\Rightarrow\mathrm{999900}\:{numbers} \\ $$$${numbers}\:{with}\:{exactly}\:{one}\:{digit}: \\ $$$${xxx},\:{xxxx},\:{xxxxx},\:{xxxxxx} \\ $$$$\Rightarrow\mathrm{4}×\mathrm{9}=\mathrm{36} \\ $$$${numbers}\:{with}\:{exactly}\:{two}\:{digits}: \\ $$$${xxy},\:{xxxy},\:{xxxxy},\:{xxxxxy} \\ $$$$\Rightarrow\left(\mathrm{3}+\mathrm{4}+\mathrm{5}+\mathrm{6}\right)×\mathrm{9}×\mathrm{9}=\mathrm{1458} \\ $$$$ \\ $$$$\mathrm{999900}−\mathrm{36}−\mathrm{1458}=\mathrm{998406}\:{numbers} \\ $$$${have}\:{at}\:{least}\:{three}\:{digits}. \\ $$
Commented by Acem last updated on 13/Nov/22
$${Sir},\:{at}\:{least}\:\mathrm{3}\:{diff}\:{dig}=\:{means}\:{three}\:{or}\:{four} \\ $$$$\:{or}\:{six}\:.\:{examples}\:\mathrm{765}\:,\:\mathrm{2365}\:, \\ $$$$\:\mathrm{22345},\:\mathrm{12345} \\ $$$$\:\mathrm{333678},\:\mathrm{339654},\:\mathrm{123456} \\ $$$$\:{If}\:{that}\:{was}\:{what}'{s}\:{meant}\:{then} \\ $$$$\:{I}\:{got}\:\mathrm{178}\:\mathrm{884} \\ $$$$ \\ $$$${If}\:{not},\:{please}\:{let}\:{me}\:{understand}\:{the}\:{question} \\ $$
Commented by mr W last updated on 13/Nov/22
$${at}\:{least}\:{three}\:\equiv\:{three}\:{or}\:{four}\:{or}\:{five} \\ $$$${or}\:{six}\:{different}\:{digits}\:{are}\:{in}\:{each}\: \\ $$$${number}. \\ $$$${e}.{g}.\:\mathrm{123123},\:\mathrm{123412},\:\mathrm{123451},\:\mathrm{123456} \\ $$$${are}\:{ok},\:{but}\:\mathrm{122222},\:\mathrm{333333}\:{not}\:{ok}. \\ $$$$\mathrm{12312},\:\mathrm{12341},\:\mathrm{12345}\:{are}\:{ok},\:{but}\:\mathrm{12112}, \\ $$$$\mathrm{11111}\:{are}\:{not}\:{ok}. \\ $$$$\mathrm{1231},\:\mathrm{1234}\:{are}\:{ok},\:{but}\:\mathrm{1212},\:\mathrm{2222}\:{not}. \\ $$$$\mathrm{123}\:{is}\:{ok},\:{but}\:\mathrm{121},\:\mathrm{333}\:{are}\:{not}\:{ok}. \\ $$$${etc}. \\ $$
Commented by Acem last updated on 14/Nov/22
$${Thank}\:{you}\:{Sir}!\:,\:{the}\:{problem}\:{is}\:{and}\:{for}\:{example}\: \\ $$$$\:{that}\:{you}\:{deem}\:{that}\:{the}\:{number}\:\mathrm{123451}\:{as} \\ $$$$\:\mathrm{5}\:{diff}.{dig}.\:{while}\:{i}\:{do}\:{it}\:{as}\:\mathrm{4}\:{diff}.\:{one}\:{with}\:\mathrm{2}\:{same} \\ $$$$\:{numbers}\:``\mathrm{1}''\:{that}\:{is}\:{why}\:{i}\:{didn}'{t}\:{compt}\:\mathrm{5}\:{diff}. \\ $$$${The}\:\mathrm{2}{nd}\:{example}\:\mathrm{123123}\:{i}\:{deem}\:{it}\:{as}\:{three} \\ $$$$\:{diff}.\:{pairs}\:\mathrm{11},\:\mathrm{22},\:\mathrm{33}\:{etc}\:{and}\:{i}\:{deem}\:{the}\:\mathrm{3}\:{diff}.\:{as} \\ $$$$\:{only}\:{XXXabc}\:{with}\:{it}\:{permutation};\:{x}\neq\:{a},\:{b}\:{nor}\:{c} \\ $$$$\:{And}\:{i}\:{agree}\:{with}\:{you}\:{in}\:{remain}\:{cases} \\ $$$$\:{Thank}\:{you}\:{bro}! \\ $$