Question Number 178021 by Acem last updated on 12/Oct/22
$$ \\ $$$${Let}\:{S}=\left\{\mathrm{1},\:\mathrm{2},\:\mathrm{3},\:\mathrm{4},\:\mathrm{5}\right\}\:,\:{we}\:{want}\:{to}\:{make}\:{a}\: \\ $$$$\:{group}\:{H}\:{of}\:{numbers}\:{have}\:{the}\:{following}\: \\ $$$$\:{properties}: \\ $$$$\:\mathrm{1}.\:{Each}\:{number}\:{has}\:{different}\:{digits}\:{and} \\ $$$$\:{taken}\:{from}\:{S} \\ $$$$\:\mathrm{2}.\:{Each}\:{number}\:{is}\:{greater}\:{than}\:\mathrm{20}\:\mathrm{000} \\ $$$$\:\mathrm{3}.\:{None}\:{of}\:{them}\:{is}\:{multiple}\:{of}\:{five} \\ $$$$ \\ $$$${How}\:{many}\:{items}\:{of}\:{H}? \\ $$
Answered by mr W last updated on 12/Oct/22
$$\mathrm{4}×\mathrm{4}×\mathrm{3}×\mathrm{2}×\mathrm{1}−\mathrm{3}×\mathrm{3}×\mathrm{2}×\mathrm{1}=\mathrm{78} \\ $$
Commented by Tawa11 last updated on 12/Oct/22
$$\mathrm{Great}\:\mathrm{sir} \\ $$
Commented by aurpeyz last updated on 12/Oct/22
$${thank}\:{you}\:{Sir}.\:{I}\:{got}\:{how}\:{you}\:{handled}\:{the} \\ $$$${first}\:{two}\:{but}\:{i}\:{didnt}\:{gwt}\:{how}\:{you}\:{did}\:{that} \\ $$$${of}\:{the}\:{multiples}\:{of}\:\mathrm{5}.\:{pls}\:{explain}\: \\ $$
Commented by mr W last updated on 12/Oct/22
$${let}'{s}\:{begin}\:{with}\:{the}\:{last}\:{digit},\:{which} \\ $$$${should}\:{not}\:{be}\:\mathrm{5}.\:{so}\:{we}\:{have}\:\mathrm{4}\:{ways}\:{to} \\ $$$${select}\:{the}\:{last}\:{digit}.\:{for}\:{the}\:{second} \\ $$$${last}\:{digit}\:{we}\:{have}\:{also}\:\mathrm{4}\:{possiblities} \\ $$$${and}\:{for}\:{the}\:{next}\:{digit}\:\mathrm{3}\:{possibilities} \\ $$$${etc}.\:{so}\:{tatally}\:{we}\:{have}\:\mathrm{4}×\mathrm{4}×\mathrm{3}×\mathrm{2}×\mathrm{1} \\ $$$${ways}.\:\:{among}\:{them}\:{some}\:{have}\:\mathrm{1}\:{as} \\ $$$${the}\:{first}\:{digit}.\:{these}\:{numbers}\:{are} \\ $$$${not}\:{valid},\:{because}\:{they}\:{are}\:{smaller} \\ $$$${than}\:\mathrm{20000}.\:{now}\:{we}\:{shoud}\:{find}\:{how} \\ $$$${mang}\:{numbers}\:{begin}\:{with}\:\mathrm{1}. \\ $$$$\mathrm{1}{abcd} \\ $$$${abcd}\:{are}\:{from}\:\mathrm{2},\mathrm{3},\mathrm{4},\mathrm{5}.\:{similary}\:{as} \\ $$$${above}\:{we}\:{can}\:{get}\:{that}\:{there}\:{are} \\ $$$$\mathrm{3}×\mathrm{3}×\mathrm{2}×\mathrm{1}\:{such}\:{numbers}.\:{so}\:{the} \\ $$$${answer}\:{is} \\ $$$$\mathrm{4}×\mathrm{4}×\mathrm{3}×\mathrm{2}×\mathrm{1}−\mathrm{3}×\mathrm{3}×\mathrm{2}×\mathrm{1}=\mathrm{78}. \\ $$
Commented by Acem last updated on 12/Oct/22
$${I}\:{appreciate}\:{the}\:{professor}'{s}\:{method},\:{it}'{s}\:{very} \\ $$$$\:{clear}\:{and}\:{easy}.\:{I}\:{was}\:{going}\:{to}\:{mention}\:{mine} \\ $$$$\:{but}\:{it}'{s}\:{a}\:{bit}\:{complicated}. \\ $$$$\boldsymbol{{Big}}\:\boldsymbol{{thanks}} \\ $$
Commented by aurpeyz last updated on 12/Oct/22
$${thank}\:{you}\:{so}\:{much} \\ $$
Commented by Acem last updated on 12/Oct/22
$$\:{A}\:{simple}\:{explanation}\:{of}\:{the}\:{almighty} \\ $$$$\:{professor}'{s}\:{method} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:{a}=\:\mathrm{20}\:\mathrm{000} \\ $$$$\:\begin{array}{|c|c|}{}&\hline{}&\hline{}&\hline{}&\hline{{Not}\:\mathrm{5}}&\hline{{NW}_{{Num}_{{s}} >{a}\:{or}\:{Num}_{{s}} <{a}} =\:\mathrm{96}}\\{\mathrm{1}}&\hline{\mathrm{2}}&\hline{\mathrm{3}}&\hline{\mathrm{4}}&\hline{\:\:\:\:\mathrm{4}}&\hline{\leftarrow\:{Start}\:{from}\:{here}}\\\hline\end{array} \\ $$$$ \\ $$$$\:\begin{array}{|c|c|}{\:\mathrm{1}}&\hline{}&\hline{}&\hline{}&\hline{{Not}\:\mathrm{5}}&\hline{{NW}_{{Num}_{{s}} <\:{a}} \:=\:\mathrm{18}}\\{−}&\hline{\mathrm{1}}&\hline{\mathrm{2}}&\hline{\mathrm{3}}&\hline{\:\:\:\:\mathrm{3}}&\hline{\leftarrow\:{Start}\:{from}\:{here}}\\\hline\end{array} \\ $$$$ \\ $$$$\:{NumItems}_{{H}} \:=\:{the}\:{difference}=\:\mathrm{78}\:{numbers} \\ $$$$ \\ $$
Commented by aurpeyz last updated on 12/Oct/22
$${nice}\:{one} \\ $$