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Question Number 125548 by Don08q last updated on 12/Dec/20

$$\mathrm{How}\mathrm{many}\mathrm{four}-\mathrm{digit}\mathrm{numbers}$$$$\mathrm{between}2000\mathrm{and}4000\mathrm{can}\mathrm{be}\mathrm{formed}$$$$\mathrm{from}\mathrm{the}\mathrm{numbers}0,1,2,3,4\mathrm{and}5$$$$\mathrm{if}\mathrm{repitition}\mathrm{is}\mathrm{allowed}?$$

Commented by liberty last updated on 13/Dec/20

$$letthenumberABCDwhere2000<ABCD<4000$$$$case\left(1\right)\underset{2,3[}{A}\underset{1,2,3,4,5]}{BCD}=2\times 5\times 6^2=360$$$$case\left(2\right)\underset{2,3}{A}\underset{0}{B}\underset{[1,2,3,4,5]}{C}D=2\times 1\times 5\times 6=60$$$$case\left(3\right)\underset{2,3}{A}\underset{0}{B}\underset{0}{C}\underset{[1,2,3,4,5]}{D}=2\times 1\times 1\times 5=10$$$$case\left(4\right)\underset{3}{A}\underset{0}{B}\underset{0}{C}\underset{0}{D}=1\times 1\times 1\times 1=1$$$$totally=431$$

Commented by bemath last updated on 12/Dec/20

$$ithinkyouranswercorrectsir$$

Answered by Olaf last updated on 12/Dec/20

$$\mathrm{The}\mathrm{first}\mathrm{digit}\mathrm{is}2\mathrm{or}3\Rightarrow 2\mathrm{possibilities}.$$$$\mathrm{For}\mathrm{the}3\mathrm{other}\mathrm{digits}\Rightarrow 6\mathrm{possibilities}.$$$$2\times 6^3=432$$$$\mathrm{Here}\mathrm{we}\mathrm{obtain}\mathrm{the}\mathrm{possibilities}$$$$\mathrm{between}2000\mathrm{and}3555.$$$$\mathrm{But}4000\mathrm{is}\mathrm{a}\mathrm{possibility}\mathrm{too}.$$$$\Rightarrow 432+1=433\mathrm{possibilities}.$$

Commented by malwaan last updated on 12/Dec/20

$$mrolaf$$$$Ithinktherightansweris$$$$432-2=430$$$$thequestionisbetween2000$$$$and4000$$$$Ifthequestionwasfrom2000$$$$to4000thentheanswerwill$$$$be432$$

Commented by malwaan last updated on 12/Dec/20

$$sorry$$$$Iamnotsure$$

Commented by mr W last updated on 12/Dec/20

$$for2000⩽x⩽4000theansweris$$$$2\times 6^3+1=433$$$$for2000<x<4000theansweris$$$$2\times 6^3-1=431$$

Commented by Don08q last updated on 12/Dec/20

$$\mathrm{Yes}.431\mathrm{is}\mathrm{the}\mathbf{correct}\mathrm{answer}.✓✓$$

Commented by Olaf last updated on 13/Dec/20

$$\mathrm{Sorry},\mathrm{my}\mathrm{english}\mathrm{is}\mathrm{not}\mathrm{fluent}.$$$$\mathrm{In}\mathrm{my}\mathrm{language},‘‘{\mathrm{between}}^{″}\mathrm{means}⩽$$

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