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Question Number 50008 by rahul 19 last updated on 13/Dec/18
$$\mathrm{5}−{digit}\:{number}\:{divisible}\:{by}\:\mathrm{3}\:{formed} \\ $$$${using}\:\mathrm{0},\mathrm{1},\mathrm{2},\mathrm{3},\mathrm{4},\mathrm{5}\:{without}\:{repetition}. \\ $$$${Total}\:{number}\:{of}\:{such}\:{no}.{s}\:{is}\:? \\ $$
Commented by rahul 19 last updated on 13/Dec/18
$$\mathrm{5}−{digit}\:{number}\:{be}\:{like}\:: \\ $$$$\:\underset{\mathrm{5}{ways}} {−}\:\underset{\mathrm{5}{ways}} {−}\:\underset{\mathrm{4}{ways}} {−}\:\underset{\mathrm{3}{ways}} {−}\:\underset{} {−} \\ $$$${Now}\:{the}\:{sum}\:{of}\:{starting}\:\mathrm{4}\:{digit}\:{can} \\ $$$${be}\:{of}\:{form}\:\mathrm{3}{k},\mathrm{3}{k}+\mathrm{1},\mathrm{3}{k}+\mathrm{2}. \\ $$$${if}\:\mathrm{3}{k}:{then}\:{last}\:{digit}\:{can}\:{be}\:\mathrm{0},\mathrm{3} \\ $$$${if}\:\mathrm{3}{k}+\mathrm{1}:{then}\:{last}\:{digit}\:{can}\:{be}\:\mathrm{2},\mathrm{5} \\ $$$${if}\:\mathrm{3}{k}+\mathrm{2}:{then}\:{last}\:{digit}\:{can}\:{be}\:\mathrm{1},\mathrm{4}. \\ $$$${Hence}\:\mathrm{2}\:{ways}\:{for}\:{last}\:{digit}. \\ $$$$\therefore\:{No}.\:{of}\:{ways}\:=\:\mathrm{5}×\mathrm{5}×\mathrm{4}×\mathrm{3}×\mathrm{2}=\mathrm{600} \\ $$$${Mrw}\:{sir},\:{pl}\:{tell}\:{the}\:{cases}\:{which}\:{i}\: \\ $$$${overcount}...{i}\:{have}\:{did}\:{like}\:{Q}.\mathrm{49408}. \\ $$$${Sir},\:{i}\:{think}\:{this}\:{method} \\ $$$$\:{is}\:\:{applicable}\:{only}\:{when}\:{repetition} \\ $$$${is}\:{allowed},{right}? \\ $$
Commented by mr W last updated on 13/Dec/18
$${you}\:{can}\:{not}\:{apply}\:{this}\:{method}\:{if} \\ $$$${no}\:{repetition}\:{is}\:{allowed}. \\ $$
Commented by rahul 19 last updated on 13/Dec/18
$${Okay}\:{Sir}. \\ $$
Commented by mr W last updated on 13/Dec/18
$${e}.{g}.\:{if}\:{the}\:\mathrm{4}−{digit}\:{number}\:{is}\:\mathrm{1230}, \\ $$$${or}\:\mathrm{2301}\:{etc}.,\:{with}\:{the}\:{remaining}\:\mathrm{4}\:{and} \\ $$$$\mathrm{5}\:{you}\:{can}\:{not}\:{build}\:{a}/{suitable}\:\:{number}. \\ $$$${if}\:{the}\:\mathrm{4}−{digit}\:{number}\:{is}\:\mathrm{1234},\:{or}\:\mathrm{3421} \\ $$$${etc}.,\:{with}\:{the}\:{remaining}\:\mathrm{0}\:{and}\:\mathrm{5}\:{you} \\ $$$${can}\:{only}\:{build}\:{one}\:{suitable}\:{number}, \\ $$$${not}\:{two}\:{as}\:{you}\:{took}.\:{i}.{e}.\:{from}\:{the} \\ $$$$\mathrm{300}\:\mathrm{4}−{digit}\:{numbers}\:{you}\:{build}, \\ $$$${many}\:{of}\:{them}\:{are}\:{invalid},\:{from}\:{each}\:{of} \\ $$$${the}\:{others}\:{you}\:{can}\:{only}\:{form}\:{one} \\ $$$${suitable}\:{number}.\:{therefore}\:{the}\:{total}\:{number} \\ $$$${must}\:{be}\:{less}\:{than}\:\mathrm{300}. \\ $$
Answered by mr W last updated on 13/Dec/18
$${to}\:{form}\:{suitable}\:{numbers}\:{we}\:{can}\:{use} \\ $$$${either}\:{digits}\:\mathrm{1}/\mathrm{2}/\mathrm{3}/\mathrm{4}/\mathrm{5}\:{or}\:\mathrm{0}/\mathrm{1}/\mathrm{2}/\mathrm{4}/\mathrm{5}. \\ $$$$\Rightarrow{total}\:{number}\:{of}\:{such}\:{numbers}\:{is} \\ $$$$\mathrm{5}!+\mathrm{4}×\mathrm{4}!=\mathrm{216} \\ $$
Commented by mr W last updated on 13/Dec/18
$${a}\:{number}\:{is}\:{divisible}\:{by}\:\mathrm{3}\:{when}\:{the} \\ $$$${sum}\:{of}\:{its}\:{digits}\:{is}\:{divisible}\:{by}\:\mathrm{3}. \\ $$$${therefore}\:{from}\:{the}\:{six}\:{given}\:{digits} \\ $$$$\mathrm{0}\:{to}\:\mathrm{5},\:{we}\:{must}\:{select}\:\mathrm{5}\:{digits}\:{whose} \\ $$$${sum}\:{is}\:{divisible}\:{by}\:\mathrm{3},\:{there}\:{are}\:{only} \\ $$$${two}\:{possibilities}: \\ $$$${either}\:\mathrm{1}/\mathrm{2}/\mathrm{3}/\mathrm{4}/\mathrm{5}\:{or}\:\mathrm{0}/\mathrm{1}/\mathrm{2}/\mathrm{4}/\mathrm{5}. \\ $$
Commented by rahul 19 last updated on 13/Dec/18
$${Very}\:{Nice}\:{Sir}! \\ $$