Let-S-1-2-3-4-5-we-want-to-make-a-group-H-of-numbers-have-the-following-properties-1-Each-number-has-different-digits-and-taken-from-S-2-Each-number-is-greater-than-20-000-3-None-

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Question Number 178021 by Acem last updated on 12/Oct/22

$$$$$$LetS=\{1,2,3,4,5\},wewanttomakea$$$$groupHofnumbershavethefollowing$$$$properties:$$$$1.Eachnumberhasdifferentdigitsand$$$$takenfromS$$$$2.Eachnumberisgreaterthan20000$$$$3.Noneofthemismultipleoffive$$$$$$$$HowmanyitemsofH?$$

Answered by mr W last updated on 12/Oct/22

$$4\times 4\times 3\times 2\times 1-3\times 3\times 2\times 1=78$$

Commented by Tawa11 last updated on 12/Oct/22

$$\mathrm{Great}\mathrm{sir}$$

Commented by aurpeyz last updated on 12/Oct/22

$$thankyouSir.Igothowyouhandledthe$$$$firsttwobutididntgwthowyoudidthat$$$$ofthemultiplesof5.plsexplain$$

Commented by mr W last updated on 12/Oct/22

$${let}^{\prime }sbeginwiththelastdigit,which$$$$shouldnotbe5.sowehave4waysto$$$$selectthelastdigit.forthesecond$$$$lastdigitwehavealso4possiblities$$$$andforthenextdigit3possibilities$$$$etc.sotatallywehave4\times 4\times 3\times 2\times 1$$$$ways.amongthemsomehave1as$$$$thefirstdigit.thesenumbersare$$$$notvalid,becausetheyaresmaller$$$$than20000.nowweshoudfindhow$$$$mangnumbersbeginwith1.$$$$1abcd$$$$abcdarefrom2,3,4,5.similaryas$$$$abovewecangetthatthereare$$$$3\times 3\times 2\times 1suchnumbers.sothe$$$$answeris$$$$4\times 4\times 3\times 2\times 1-3\times 3\times 2\times 1=78.$$

Commented by Acem last updated on 12/Oct/22

$$Iappreciatethe{professor}^{\prime }smethod,{it}^{\prime }svery$$$$clearandeasy.Iwasgoingtomentionmine$$$$but{it}^{\prime }sabitcomplicated.$$$$\mathit{B}\mathit{i}\mathit{g}\mathit{t}\mathit{h}\mathit{a}\mathit{n}\mathit{k}\mathit{s}$$

Commented by aurpeyz last updated on 12/Oct/22

$$thankyousomuch$$

Commented by Acem last updated on 12/Oct/22

$$Asimpleexplanationofthealmighty$$$${professor}^{\prime }smethod$$$$a=20000$$$$\overline{)\begin{array}{cccccc}& & & & Not5& {NW}_{{Num}_s>aor{Num}_s<a}=96\\ 1& 2& 3& 4& 4& \leftarrow Startfromhere\end{array}}$$$$$$$$\overline{)\begin{array}{cccccc}1& & & & Not5& {NW}_{{Num}_s<a}=18\\ -& 1& 2& 3& 3& \leftarrow Startfromhere\end{array}}$$$$$$$${NumItems}_H=thedifference=78numbers$$$$$$

Commented by aurpeyz last updated on 12/Oct/22

$$niceone$$

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