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Question Number 178395 by mr W last updated on 16/Oct/22
How many 5 digit numbers with different digits are multiple of 9?
$${How}\:{many}\:\mathrm{5}\:{digit}\:{numbers}\:{with} \\ $$$${different}\:{digits}\:{are}\:{multiple}\:{of}\:\mathrm{9}? \\ $$
Answered by cortano1 last updated on 16/Oct/22
(1)1+5+6+7+8⇒5! (2)1+2+3+4+8⇒5! (3)1+2+4+5+6⇒5! (4)1+2+3+5+7⇒5! (5)1+3+6+8+9⇒5! (6)1+4+5+8+9⇒5! (7)1+2+7+8+9⇒5! (8)2+3+5+8+9⇒5! ⇒8×5!=8×120=960
$$\left(\mathrm{1}\right)\mathrm{1}+\mathrm{5}+\mathrm{6}+\mathrm{7}+\mathrm{8}\Rightarrow\mathrm{5}! \\ $$$$\left(\mathrm{2}\right)\mathrm{1}+\mathrm{2}+\mathrm{3}+\mathrm{4}+\mathrm{8}\Rightarrow\mathrm{5}! \\ $$$$\left(\mathrm{3}\right)\mathrm{1}+\mathrm{2}+\mathrm{4}+\mathrm{5}+\mathrm{6}\Rightarrow\mathrm{5}! \\ $$$$\left(\mathrm{4}\right)\mathrm{1}+\mathrm{2}+\mathrm{3}+\mathrm{5}+\mathrm{7}\Rightarrow\mathrm{5}! \\ $$$$\left(\mathrm{5}\right)\mathrm{1}+\mathrm{3}+\mathrm{6}+\mathrm{8}+\mathrm{9}\Rightarrow\mathrm{5}! \\ $$$$\left(\mathrm{6}\right)\mathrm{1}+\mathrm{4}+\mathrm{5}+\mathrm{8}+\mathrm{9}\Rightarrow\mathrm{5}! \\ $$$$\left(\mathrm{7}\right)\mathrm{1}+\mathrm{2}+\mathrm{7}+\mathrm{8}+\mathrm{9}\Rightarrow\mathrm{5}! \\ $$$$\left(\mathrm{8}\right)\mathrm{2}+\mathrm{3}+\mathrm{5}+\mathrm{8}+\mathrm{9}\Rightarrow\mathrm{5}! \\ $$$$\Rightarrow\mathrm{8}×\mathrm{5}!=\mathrm{8}×\mathrm{120}=\mathrm{960} \\ $$$$ \\ $$
Commented by mr W last updated on 16/Oct/22
what about numbers like 13068, 47981 etc.?
$${what}\:{about}\:{numbers}\:{like}\:\mathrm{13068},\:\mathrm{47981}\: \\ $$$${etc}.? \\ $$
Commented by Tawa11 last updated on 16/Oct/22
Great sir
$$\mathrm{Great}\:\mathrm{sir} \\ $$
Commented by mr W last updated on 16/Oct/22
i meant in his solution numbers like 13068, 47981 etc. are not included.
$${i}\:{meant}\:{in}\:{his}\:{solution}\:{numbers}\:{like} \\ $$$$\mathrm{13068},\:\mathrm{47981}\:{etc}.\:{are}\:{not}\:{included}. \\ $$
Commented by mr W last updated on 16/Oct/22
so answer 960 is not correct.
$${so}\:{answer}\:\mathrm{960}\:{is}\:{not}\:{correct}. \\ $$
Commented by cortano1 last updated on 16/Oct/22
do you meant the numbers is formed from 0,1,...,9?
$$\mathrm{do}\:\mathrm{you}\:\mathrm{meant}\:\mathrm{the}\:\mathrm{numbers}\:\mathrm{is} \\ $$$$\mathrm{formed}\:\mathrm{from}\:\mathrm{0},\mathrm{1},…,\mathrm{9}? \\ $$
Commented by mr W last updated on 16/Oct/22
yes. all digits are allowed.
$${yes}.\:{all}\:{digits}\:{are}\:{allowed}. \\ $$
Commented by cortano1 last updated on 16/Oct/22
ok sir
$$\mathrm{ok}\:\mathrm{sir} \\ $$
Commented by mr W last updated on 16/Oct/22
typo. it′s just an example.
$${typo}.\:{it}'{s}\:{just}\:{an}\:{example}. \\ $$
Commented by cortano1 last updated on 16/Oct/22
what your answer sir?
$$\mathrm{what}\:\mathrm{your}\:\mathrm{answer}\:\mathrm{sir}? \\ $$
Commented by mr W last updated on 16/Oct/22
i got 3024.
$${i}\:{got}\:\mathrm{3024}. \\ $$
Answered by mr W last updated on 16/Oct/22
0+1+2+3+4=10 9+8+7+6+5=35 that means the sum of 5 different digits lies between 10 and 35. such that a 5 digit number is multiple of 9, the sum of its digits must be 27 or 18. case 1: sum of digits=27 9+8+7+3+0 ⇒(4/5)×5! numbers 9+8+6+4+0 9+7+6+5+0 9+8+7+2+1 ⇒5! numbers 9+8+6+3+1 9+8+5+4+1 9+8+5+3+2 9+7+6+4+1 9+7+6+3+2 9+7+5+4+2 9+6+5+4+3 8+7+6+5+1 8+7+6+4+2 8+7+5+4+3 ⇒3×(4/5)×5!+11×5! case 2: sum of digits=18 9+6+2+1+0 ⇒(4/5)×5! numbers 9+5+3+1+0 9+4+3+2+0 8+7+2+1+0 8+6+3+1+0 8+5+4+1+0 8+5+3+2+0 7+6+4+1+0 7+6+3+2+0 7+5+4+2+0 6+5+4+3+0 8+4+3+2+1 ⇒5! numbers 7+5+3+2+1 6+5+4+2+1 ⇒11×(4/5)×5!+3×5! totally: ⇒(11+3)(1+(4/5))×5!=14×9×4!=3024
$$\mathrm{0}+\mathrm{1}+\mathrm{2}+\mathrm{3}+\mathrm{4}=\mathrm{10} \\ $$$$\mathrm{9}+\mathrm{8}+\mathrm{7}+\mathrm{6}+\mathrm{5}=\mathrm{35} \\ $$$${that}\:{means}\:{the}\:{sum}\:{of}\:\mathrm{5}\:{different} \\ $$$${digits}\:{lies}\:{between}\:\mathrm{10}\:{and}\:\mathrm{35}.\: \\ $$$${such}\:{that}\:{a}\:\mathrm{5}\:{digit}\:{number}\:{is}\: \\ $$$${multiple}\:{of}\:\mathrm{9},\:{the}\:{sum}\:{of}\:{its}\:{digits}\: \\ $$$${must}\:{be}\:\mathrm{27}\:{or}\:\mathrm{18}. \\ $$$$ \\ $$$$\boldsymbol{{case}}\:\mathrm{1}:\:\boldsymbol{{sum}}\:\boldsymbol{{of}}\:\boldsymbol{{digits}}=\mathrm{27} \\ $$$$\mathrm{9}+\mathrm{8}+\mathrm{7}+\mathrm{3}+\mathrm{0}\:\Rightarrow\frac{\mathrm{4}}{\mathrm{5}}×\mathrm{5}!\:{numbers} \\ $$$$\mathrm{9}+\mathrm{8}+\mathrm{6}+\mathrm{4}+\mathrm{0} \\ $$$$\mathrm{9}+\mathrm{7}+\mathrm{6}+\mathrm{5}+\mathrm{0} \\ $$$$\mathrm{9}+\mathrm{8}+\mathrm{7}+\mathrm{2}+\mathrm{1}\:\Rightarrow\mathrm{5}!\:{numbers} \\ $$$$\mathrm{9}+\mathrm{8}+\mathrm{6}+\mathrm{3}+\mathrm{1} \\ $$$$\mathrm{9}+\mathrm{8}+\mathrm{5}+\mathrm{4}+\mathrm{1} \\ $$$$\mathrm{9}+\mathrm{8}+\mathrm{5}+\mathrm{3}+\mathrm{2} \\ $$$$\mathrm{9}+\mathrm{7}+\mathrm{6}+\mathrm{4}+\mathrm{1} \\ $$$$\mathrm{9}+\mathrm{7}+\mathrm{6}+\mathrm{3}+\mathrm{2} \\ $$$$\mathrm{9}+\mathrm{7}+\mathrm{5}+\mathrm{4}+\mathrm{2} \\ $$$$\mathrm{9}+\mathrm{6}+\mathrm{5}+\mathrm{4}+\mathrm{3} \\ $$$$\mathrm{8}+\mathrm{7}+\mathrm{6}+\mathrm{5}+\mathrm{1} \\ $$$$\mathrm{8}+\mathrm{7}+\mathrm{6}+\mathrm{4}+\mathrm{2} \\ $$$$\mathrm{8}+\mathrm{7}+\mathrm{5}+\mathrm{4}+\mathrm{3}\: \\ $$$$\Rightarrow\mathrm{3}×\frac{\mathrm{4}}{\mathrm{5}}×\mathrm{5}!+\mathrm{11}×\mathrm{5}! \\ $$$$ \\ $$$$\boldsymbol{{case}}\:\mathrm{2}:\:\boldsymbol{{sum}}\:\boldsymbol{{of}}\:\boldsymbol{{digits}}=\mathrm{18} \\ $$$$\mathrm{9}+\mathrm{6}+\mathrm{2}+\mathrm{1}+\mathrm{0}\:\Rightarrow\frac{\mathrm{4}}{\mathrm{5}}×\mathrm{5}!\:{numbers} \\ $$$$\mathrm{9}+\mathrm{5}+\mathrm{3}+\mathrm{1}+\mathrm{0} \\ $$$$\mathrm{9}+\mathrm{4}+\mathrm{3}+\mathrm{2}+\mathrm{0} \\ $$$$\mathrm{8}+\mathrm{7}+\mathrm{2}+\mathrm{1}+\mathrm{0} \\ $$$$\mathrm{8}+\mathrm{6}+\mathrm{3}+\mathrm{1}+\mathrm{0} \\ $$$$\mathrm{8}+\mathrm{5}+\mathrm{4}+\mathrm{1}+\mathrm{0} \\ $$$$\mathrm{8}+\mathrm{5}+\mathrm{3}+\mathrm{2}+\mathrm{0} \\ $$$$\mathrm{7}+\mathrm{6}+\mathrm{4}+\mathrm{1}+\mathrm{0} \\ $$$$\mathrm{7}+\mathrm{6}+\mathrm{3}+\mathrm{2}+\mathrm{0} \\ $$$$\mathrm{7}+\mathrm{5}+\mathrm{4}+\mathrm{2}+\mathrm{0} \\ $$$$\mathrm{6}+\mathrm{5}+\mathrm{4}+\mathrm{3}+\mathrm{0} \\ $$$$\mathrm{8}+\mathrm{4}+\mathrm{3}+\mathrm{2}+\mathrm{1}\:\Rightarrow\mathrm{5}!\:{numbers} \\ $$$$\mathrm{7}+\mathrm{5}+\mathrm{3}+\mathrm{2}+\mathrm{1} \\ $$$$\mathrm{6}+\mathrm{5}+\mathrm{4}+\mathrm{2}+\mathrm{1} \\ $$$$\Rightarrow\mathrm{11}×\frac{\mathrm{4}}{\mathrm{5}}×\mathrm{5}!+\mathrm{3}×\mathrm{5}! \\ $$$$ \\ $$$$\boldsymbol{{totally}}: \\ $$$$\Rightarrow\left(\mathrm{11}+\mathrm{3}\right)\left(\mathrm{1}+\frac{\mathrm{4}}{\mathrm{5}}\right)×\mathrm{5}!=\mathrm{14}×\mathrm{9}×\mathrm{4}!=\mathrm{3024} \\ $$
Commented by SLVR last updated on 25/Oct/22
it is more clear...awesom.super
$${it}\:{is}\:{more}\:{clear}…{awesom}.{super} \\ $$

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