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Question Number 51287 by aseerimad last updated on 25/Dec/18
Answered by mr W last updated on 26/Dec/18
$${let}'{s}\:{look}\:{at}\:{an}\:{example}: \\ $$$${the}\:{number}\:\mathrm{1234}\:{consists}\:{of}\:\mathrm{4}\:{digits}: \\ $$$$\mathrm{1}\:{represents}\:{here}\:{the}\:{value}\:\mathrm{1}×\mathrm{1000} \\ $$$$\mathrm{2}\:{represents}\:{here}\:{the}\:{value}\:\mathrm{2}×\mathrm{100} \\ $$$$\mathrm{3}\:{represents}\:{here}\:{the}\:{value}\:\mathrm{3}×\mathrm{10} \\ $$$$\mathrm{4}\:{represents}\:{here}\:{the}\:{value}\:\mathrm{4}×\mathrm{1} \\ $$$$\Rightarrow\mathrm{1234}=\mathrm{1}×\mathrm{1000}+\mathrm{2}×\mathrm{100}+\mathrm{3}×\mathrm{10}+\mathrm{4}×\mathrm{1} \\ $$$${so}\:{when}\:{we}\:{want}\:{to}\:{get}\:{the}\:{sum}\:{of}\:{some} \\ $$$${numbers},\:{we}\:{only}\:{need}\:{to}\:{calculate} \\ $$$${the}\:{sum}\:{of}\:{the}\:{values}\:{which}\:{their} \\ $$$${digits}\:{represent}.\:{for}\:{example}: \\ $$$$\mathrm{1234}+\mathrm{4321} \\ $$$$=\left(\mathrm{1}+\mathrm{4}\right)×\mathrm{1000}+\left(\mathrm{2}+\mathrm{3}\right)×\mathrm{100}+\left(\mathrm{3}+\mathrm{2}\right)×\mathrm{10}+\left(\mathrm{4}+\mathrm{1}\right)×\mathrm{1} \\ $$$$=\mathrm{5}×\left(\mathrm{1000}+\mathrm{100}+\mathrm{10}+\mathrm{1}\right) \\ $$$$=\mathrm{5}×\mathrm{1111} \\ $$$$=\mathrm{5555} \\ $$$${we}\:{can}\:{use}\:{this}\:{method}\:{to}\:{solve}\:{our} \\ $$$${problem}.\:{its}\:{advantage}\:{is}\:{that}\:{we} \\ $$$${don}'{t}\:{need}\:{to}\:{know}\:{the}\:{numbers}\: \\ $$$${themself}.\:{we}\:{only}\:{need}\:{to}\:{know}\:{the} \\ $$$${digits}\:{from}\:{which}\:{the}\:{numbers}\:{are} \\ $$$${formed}.\:{and}\:{we}\:{know}\:{the}\:{digits},\:{they} \\ $$$${are}\:\mathrm{1},\mathrm{2},\mathrm{3}\:{and}\:\mathrm{4}. \\ $$$$ \\ $$$${let}'{s}\:{look}\:{at}\:{the}\:{digit}\:\mathrm{1}:\: \\ $$$${it}\:{can}\:{be}\:{at}\:{the}\:{thousands}\:{place}\:{as} \\ $$$$\mathrm{1}{XYZ}.\:{we}\:{know}\:{there}\:{are}\:\mathrm{3}!\:{numbers} \\ $$$${in}\:{which}\:{the}\:{digit}\:\mathrm{1}\:{is}\:{at}\:{the}\:{thousands} \\ $$$${place}.\:{the}\:{sum}\:{of}\:{all}\:{values}\:{which}\:{the} \\ $$$${digit}\:\mathrm{1}\:{represent}\:{in}\:{the}\:{thousands}\:{place} \\ $$$${is}\:{then}\:\mathrm{1}×\mathrm{1000}×\mathrm{3}!. \\ $$$${similarly}\:{the}\:{digit}\:\mathrm{1}\:{can}\:{be}\:{in}\:{the} \\ $$$${hundreds}\:{place}.\:{there}\:{are}\:{also}\:\mathrm{3}!\:{such} \\ $$$${numbers}.\:{the}\:{sum}\:{of}\:{all}\:{values}\:{which}\:{the} \\ $$$${digit}\:\mathrm{1}\:{represent}\:{in}\:{the}\:{hundreds}\:{place} \\ $$$${is}\:\mathrm{1}×\mathrm{100}×\mathrm{3}!. \\ $$$${similarly}\:{the}\:{digit}\:\mathrm{1}\:{can}\:{be}\:{in}\:{the} \\ $$$${tens}\:{place}.\:{there}\:{are}\:{also}\:\mathrm{3}!\:{such} \\ $$$${numbers}.\:{the}\:{sum}\:{of}\:{all}\:{values}\:{which}\:{the} \\ $$$${digit}\:\mathrm{1}\:{represent}\:{in}\:{the}\:{tens}\:{place} \\ $$$${is}\:\mathrm{1}×\mathrm{10}×\mathrm{3}!. \\ $$$${finally}\:{the}\:{digit}\:\mathrm{1}\:{can}\:{be}\:{in}\:{the} \\ $$$${units}\:{place}.\:{there}\:{are}\:{also}\:\mathrm{3}!\:{such} \\ $$$${numbers}.\:{the}\:{sum}\:{of}\:{all}\:{values}\:{which}\:{the} \\ $$$${digit}\:\mathrm{1}\:{represent}\:{in}\:{the}\:{units}\:{place} \\ $$$${is}\:\mathrm{1}×\mathrm{1}×\mathrm{3}!. \\ $$$${in}\:{this}\:{way}\:{we}\:{get}\:{the}\:{sum}\:{of}\:{all} \\ $$$${values}\:{which}\:{the}\:{digit}\:\mathrm{1}\:{represent}\:{in} \\ $$$${all}\:{possible}\:{places}\:{is}: \\ $$$$\mathrm{1}×\left(\mathrm{1000}+\mathrm{100}+\mathrm{10}+\mathrm{1}\right)×\mathrm{3}! \\ $$$${this}\:{applies}\:{also}\:{for}\:{the}\:{digits}\:\mathrm{2},\mathrm{3},\mathrm{4}. \\ $$$$ \\ $$$${the}\:{sum}\:{of}\:{all}\:{values}\:{which}\:{all}\:{of} \\ $$$${these}\:\mathrm{4}\:{digits}\:{in}\:{all}\:{possible}\:{places} \\ $$$${represent}\:{is}\:{then} \\ $$$$\left(\mathrm{1}+\mathrm{2}+\mathrm{3}+\mathrm{4}\right)×\left(\mathrm{1000}+\mathrm{100}+\mathrm{10}+\mathrm{1}\right)×\mathrm{3}! \\ $$$${this}\:{is}\:{also}\:{the}\:{sum}\:{of}\:{all}\:{numbers} \\ $$$${formed}\:{by}\:{these}\:\mathrm{4}\:{digits}. \\ $$$$ \\ $$$${so}\:{the}\:{answer}\:{for}\:{the}\:{question}\:{is} \\ $$$$\left(\mathrm{1}+\mathrm{2}+\mathrm{3}+\mathrm{4}\right)×\left(\mathrm{1000}+\mathrm{100}+\mathrm{10}+\mathrm{1}\right)×\mathrm{3}! \\ $$$$=\mathrm{10}×\left(\mathrm{1000}+\mathrm{100}+\mathrm{10}+\mathrm{1}\right)×\mathrm{3}! \\ $$$$=\mathrm{10}×\mathrm{1111}×\mathrm{3}! \\ $$$$=\mathrm{10}×\mathrm{6666} \\ $$$$=\mathrm{66660} \\ $$
Commented by aseerimad last updated on 26/Dec/18
$${can}\:{you}\:{pls}\:{explain}\:{the}\:{method}? \\ $$
Commented by mr W last updated on 26/Dec/18
$${i}\:{have}\:{added}\:{some}\:{explanation}\:{into} \\ $$$${the}\:{workings}.\:{hope}\:{it}'{s}\:{clear}\:{now}. \\ $$
Commented by aseerimad last updated on 26/Dec/18
$${Thank}\:{you}\:{Thank}\:{you}\:{very}\:{much}.{Thank}\:{you}\:{for}\:{explaining}\:{in}\:{such}\:{a}\:{lucid}\:{way}. \\ $$$${God}\:{Bless}\:{you}. \\ $$