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

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