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Question Number 180200 by Acem last updated on 09/Nov/22
$$\left.\boldsymbol{{A}}\right)\:{How}\:{many}\:{even}\:{numbers}\:{of}\:\mathrm{3}\:{different}\:{digits} \\ $$$$\:{bigger}\:{than}\:\mathrm{300}\:{can}\:{be}\:{formed}\:{from}\:\left\{\mathrm{1},\:\mathrm{2},\:\mathrm{3},\:\mathrm{4}\right\} \\ $$$$\left.\boldsymbol{{B}}\right)\:{How}\:{many}\:{numbers}\:{bigger}\:{than}\:\mathrm{300}\:{can} \\ $$$$\:\:\:\:\:\:\:\:{be}\:{formed}\:{from}\:{the}\:{same}\:{group} \\ $$
Commented by Frix last updated on 09/Nov/22
$$\mathrm{6} \\ $$
Commented by Acem last updated on 09/Nov/22
$${Right}!\:{thank}\:{you},\:\:{well},\:{please},\:{put}\:{the}\:{solution} \\ $$$$\:{method}\:{in}\:{answers}\:{part} \\ $$
Answered by Acem last updated on 09/Nov/22
Commented by Acem last updated on 09/Nov/22
$$\left.{A}\right)\:{Num}_{{Numb}.} =\:\mathrm{6} \\ $$$$\left.{B}\right)\:{Num}_{{Numb}.} =\:\cancel{\mathrm{288}}\:\:{See}\:{Mr}.\:{W}\:'{s}\:{note}\:{above} \\ $$
Commented by Acem last updated on 09/Nov/22
$${And}\:{yes}\:{I}\:{agrre}\:{with}\:{Mr}.\:{W}\:{that}\:{there}\:{are}\:{infinite} \\ $$$$\:{numbers}\:{of}\:{infinite}\:{digits}! \\ $$
Answered by Frix last updated on 09/Nov/22
$$\mathrm{312}\:\mathrm{314}\:\mathrm{324}\:\mathrm{342}\:\mathrm{412}\:\mathrm{432} \\ $$$$\mathrm{6}\:\mathrm{numbers} \\ $$
Commented by Acem last updated on 09/Nov/22
$${Good},\:{and}\:{it}'{s}\:{better}\:{is}\:{forming}\:{a}\:{tree}\:{like}\:{this}: \\ $$$$ \\ $$
Commented by Acem last updated on 09/Nov/22
Commented by Acem last updated on 09/Nov/22
$${But}\:{what}\:{if}\:{the}\:{group}\:{S}\:{had}\:{items}\:{bigger}\:{than}\:\mathrm{4}? \\ $$$$\:{like}\:\mathrm{6},\:\mathrm{7},...,\:\mathrm{10}\:{numbers}. \\ $$$$\:{Or}\:{the}\:{numbers}\:{were}\:{of}\:\mathrm{9}\:{digits}? \\ $$$$\:{Even}\:{the}\:{tree}\:{method}\:{would}\:{be}\:{complex} \\ $$$$\left.\:{well}\:{try}\:{to}\:{find}\:{other}\:{way}\:{to}\:{solve}\:{it}\:\::\right) \\ $$$$\: \\ $$
Commented by Acem last updated on 09/Nov/22
$$\:{Note}:\:{A}\:{new}\:{question}\:{has}\:{been}\:{added} \\ $$
Answered by Rasheed.Sindhi last updated on 09/Nov/22
$$\left.\boldsymbol{{B}}\right)\:\:\:\:\overline {{abc}} \\ $$$$\:\:\:\:\:\:{a}=\mathrm{3}\:{or}\:\mathrm{4}:\:\mathrm{2}\:{ways} \\ $$$$\:\:\:\:\:\:{b}:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\mathrm{4}\:{ways} \\ $$$$\:\:\:\:\:\:{c}:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\mathrm{4}\:{ways} \\ $$$$\mathcal{T}{otal}\:{ways}\:\:\:\:\mathrm{2}\centerdot\mathrm{4}\centerdot\mathrm{4}=\mathrm{32}\: \\ $$
Commented by Acem last updated on 09/Nov/22
$$\left.{Sorry}\:\right):\:\:{try}\:{again}...{you}\:{will}\:{do}\:{it}! \\ $$
Commented by Acem last updated on 09/Nov/22
$$\:{There}'{s}\:{a}\:{trap}\:{in}\:{my}\:\mathrm{2}{nd}\:{question} \\ $$
Answered by mr W last updated on 09/Nov/22
$$\left.{A}\right) \\ $$$$\mathrm{2}×\mathrm{3}×\mathrm{2}=\mathrm{12}\:{numbers}: \\ $$$$\mathrm{342} \\ $$$$\mathrm{341} \\ $$$$\mathrm{324} \\ $$$$\mathrm{321} \\ $$$$\mathrm{314} \\ $$$$\mathrm{312} \\ $$$$\mathrm{432} \\ $$$$\mathrm{431} \\ $$$$\mathrm{423} \\ $$$$\mathrm{421} \\ $$$$\mathrm{413} \\ $$$$\mathrm{412} \\ $$$$ \\ $$$$\left.{B}\right) \\ $$$${assume}\:{different}\:{digits},\:{then} \\ $$$$\mathrm{3}−{digit}\:{numbers}:\:\mathrm{12}\:{numbers} \\ $$$$\mathrm{4}−{digit}\:{numbers}:\:\mathrm{4}!=\mathrm{24}\:{numbers} \\ $$$${totally}:\:\mathrm{36}\:{numbers} \\ $$$$ \\ $$$${if}\:{digits}\:{may}\:{repeat},\:{then}\:{we}\:{can} \\ $$$${form}\:{infinite}\:{numbers}\:{larger}\:{than} \\ $$$$\mathrm{300}. \\ $$
Commented by JDamian last updated on 09/Nov/22
Mr. W, the question A says *even* numbers.
Commented by mr W last updated on 09/Nov/22
$${sorry}\:{i}\:{overlooked}\:{that}.\:{thanks}! \\ $$
Commented by Acem last updated on 09/Nov/22
$${Yes}\:{Sir}!\:{thanks}\:\left(:\right. \\ $$