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13456622577532674-how-many-5-digit-numbers-can-be-made-from-these-numbers-help-please-

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Question Number 209794 by Ismoiljon_008 last updated on 21/Jul/24
13456622577532674 how many 5-digit numbers can be made from these numbers? help please
$$\:\:\: \\ $$$$\:\:\:\mathrm{13456622577532674}\:{how}\:{many}\:\mathrm{5}-{digit} \\ $$$$\:\:\:{numbers}\:{can}\:{be}\:{made}\:{from}\:{these}\:{numbers}? \\ $$$$\:\:\:{help}\:{please} \\ $$$$ \\ $$
Answered by MM42 last updated on 21/Jul/24
1→1 / 2→3 / 3→2 / 4→2/ 5→3 /6→3 / 7→3 abcde⇒7×6×5×4×3=2520 aabcd⇒ ((6),(1) )× ((6),(3) )×((5!)/(2!))=7200 aabbc⇒ ((6),(2) )× ((5),(1) )×((5!)/(2!2!))=2250 aaabc⇒ ((4),(1) )× ((6),(2) )×((5!)/(3!))=1200 aaabb⇒ ((4),(1) )× ((5),(1) )×((5!)/(3!2!))=200 ans=13370 ✓
$$\mathrm{1}\rightarrow\mathrm{1}\:\:/\:\:\mathrm{2}\rightarrow\mathrm{3}\:\:/\:\:\mathrm{3}\rightarrow\mathrm{2}\:/\:\:\mathrm{4}\rightarrow\mathrm{2}/\:\mathrm{5}\rightarrow\mathrm{3}\:/\mathrm{6}\rightarrow\mathrm{3}\:/\:\mathrm{7}\rightarrow\mathrm{3}\: \\ $$$${abcde}\Rightarrow\mathrm{7}×\mathrm{6}×\mathrm{5}×\mathrm{4}×\mathrm{3}=\mathrm{2520} \\ $$$${aabcd}\Rightarrow\begin{pmatrix}{\mathrm{6}}\\{\mathrm{1}}\end{pmatrix}×\begin{pmatrix}{\mathrm{6}}\\{\mathrm{3}}\end{pmatrix}×\frac{\mathrm{5}!}{\mathrm{2}!}=\mathrm{7200} \\ $$$${aabbc}\Rightarrow\begin{pmatrix}{\mathrm{6}}\\{\mathrm{2}}\end{pmatrix}×\begin{pmatrix}{\mathrm{5}}\\{\mathrm{1}}\end{pmatrix}×\frac{\mathrm{5}!}{\mathrm{2}!\mathrm{2}!}=\mathrm{2250} \\ $$$${aaabc}\Rightarrow\begin{pmatrix}{\mathrm{4}}\\{\mathrm{1}}\end{pmatrix}×\begin{pmatrix}{\mathrm{6}}\\{\mathrm{2}}\end{pmatrix}×\frac{\mathrm{5}!}{\mathrm{3}!}=\mathrm{1200} \\ $$$${aaabb}\Rightarrow\begin{pmatrix}{\mathrm{4}}\\{\mathrm{1}}\end{pmatrix}×\begin{pmatrix}{\mathrm{5}}\\{\mathrm{1}}\end{pmatrix}×\frac{\mathrm{5}!}{\mathrm{3}!\mathrm{2}!}=\mathrm{200} \\ $$$${ans}=\mathrm{13370}\:\checkmark \\ $$$$ \\ $$
Commented by Ismoiljon_008 last updated on 22/Jul/24
thank you very much
$$\:\:\:{thank}\:{you}\:{very}\:{much} \\ $$
Commented by mr W last updated on 22/Jul/24
method is right, but answer is wrong. abcde⇒7×6×5×4×3=2520 ✓ aabcd⇒ ((5),(1) )× ((6),(3) )×((5!)/(2!))=6000 aabbc⇒ ((5),(2) )× ((5),(1) )×((5!)/(2!2!))=1500 aaabc⇒ ((4),(1) )× ((6),(2) )×((5!)/(3!))=1200 ✓ aaabb⇒ ((4),(1) )× ((4),(1) )×((5!)/(3!2!))=160 totally: 11380
$${method}\:{is}\:{right},\:{but}\:{answer}\:{is}\:{wrong}. \\ $$$${abcde}\Rightarrow\mathrm{7}×\mathrm{6}×\mathrm{5}×\mathrm{4}×\mathrm{3}=\mathrm{2520}\:\checkmark \\ $$$${aabcd}\Rightarrow\begin{pmatrix}{\mathrm{5}}\\{\mathrm{1}}\end{pmatrix}×\begin{pmatrix}{\mathrm{6}}\\{\mathrm{3}}\end{pmatrix}×\frac{\mathrm{5}!}{\mathrm{2}!}=\mathrm{6000} \\ $$$${aabbc}\Rightarrow\begin{pmatrix}{\mathrm{5}}\\{\mathrm{2}}\end{pmatrix}×\begin{pmatrix}{\mathrm{5}}\\{\mathrm{1}}\end{pmatrix}×\frac{\mathrm{5}!}{\mathrm{2}!\mathrm{2}!}=\mathrm{1500} \\ $$$${aaabc}\Rightarrow\begin{pmatrix}{\mathrm{4}}\\{\mathrm{1}}\end{pmatrix}×\begin{pmatrix}{\mathrm{6}}\\{\mathrm{2}}\end{pmatrix}×\frac{\mathrm{5}!}{\mathrm{3}!}=\mathrm{1200}\:\checkmark \\ $$$${aaabb}\Rightarrow\begin{pmatrix}{\mathrm{4}}\\{\mathrm{1}}\end{pmatrix}×\begin{pmatrix}{\mathrm{4}}\\{\mathrm{1}}\end{pmatrix}×\frac{\mathrm{5}!}{\mathrm{3}!\mathrm{2}!}=\mathrm{160} \\ $$$${totally}:\:\mathrm{11380} \\ $$
Answered by mr W last updated on 22/Jul/24
an other path using generating function: 1, 3 44 222, 555, 666, 777 the answer is the coef. of term x^5 of the expansion 5!(1+x)^2 (1+x+(x^2 /(2!)))(1+x+(x^2 /(2!))+(x^3 /(3!)))^4 which is 11380.
$${an}\:{other}\:{path}\:{using}\:{generating}\:{function}: \\ $$$$\mathrm{1},\:\mathrm{3} \\ $$$$\mathrm{44} \\ $$$$\mathrm{222},\:\mathrm{555},\:\mathrm{666},\:\mathrm{777} \\ $$$${the}\:{answer}\:{is}\:{the}\:{coef}.\:{of}\:{term}\:{x}^{\mathrm{5}} \:{of} \\ $$$${the}\:{expansion} \\ $$$$\mathrm{5}!\left(\mathrm{1}+{x}\right)^{\mathrm{2}} \left(\mathrm{1}+{x}+\frac{{x}^{\mathrm{2}} }{\mathrm{2}!}\right)\left(\mathrm{1}+{x}+\frac{{x}^{\mathrm{2}} }{\mathrm{2}!}+\frac{{x}^{\mathrm{3}} }{\mathrm{3}!}\right)^{\mathrm{4}} \\ $$$${which}\:{is}\:\mathrm{11380}. \\ $$
Commented by mr W last updated on 22/Jul/24
Commented by mr W last updated on 22/Jul/24
if the question is “how many 7 digit numbers can be made”, then the answer is the coef. of term x^7 of 7!(1+x)^2 (1+x+(x^2 /(2!)))(1+x+(x^2 /(2!))+(x^3 /(3!)))^4 which is 358680.
$${if}\:{the}\:{question}\:{is}\:“{how}\:{many}\:\mathrm{7} \\ $$$${digit}\:{numbers}\:{can}\:{be}\:{made}'',\:{then} \\ $$$${the}\:{answer}\:{is}\:{the}\:{coef}.\:{of}\:{term}\:{x}^{\mathrm{7}} \:{of} \\ $$$$\mathrm{7}!\left(\mathrm{1}+{x}\right)^{\mathrm{2}} \left(\mathrm{1}+{x}+\frac{{x}^{\mathrm{2}} }{\mathrm{2}!}\right)\left(\mathrm{1}+{x}+\frac{{x}^{\mathrm{2}} }{\mathrm{2}!}+\frac{{x}^{\mathrm{3}} }{\mathrm{3}!}\right)^{\mathrm{4}} \\ $$$${which}\:{is}\:\mathrm{358680}. \\ $$
Commented by mr W last updated on 22/Jul/24
Commented by mr W last updated on 22/Jul/24
or we just consider the generating function (1+x)^2 (1+x+(x^2 /(2!)))(1+x+(x^2 /(2!))+(x^3 /(3!)))^4 then the number of k−digit numbers is k!×coefficient of term x^k .
$${or}\:{we}\:{just}\:{consider}\:{the}\:{generating} \\ $$$${function} \\ $$$$\left(\mathrm{1}+{x}\right)^{\mathrm{2}} \left(\mathrm{1}+{x}+\frac{{x}^{\mathrm{2}} }{\mathrm{2}!}\right)\left(\mathrm{1}+{x}+\frac{{x}^{\mathrm{2}} }{\mathrm{2}!}+\frac{{x}^{\mathrm{3}} }{\mathrm{3}!}\right)^{\mathrm{4}} \\ $$$${then}\:{the}\:{number}\:{of}\:{k}−{digit}\:{numbers} \\ $$$${is}\:{k}!×{coefficient}\:{of}\:{term}\:{x}^{{k}} . \\ $$
Commented by mr W last updated on 22/Jul/24
Commented by mr W last updated on 22/Jul/24
examples: 4 digit numbers: 4!×79=1896 5 digit numbers: 5!×((569)/6)=11380 6 digit numbers: 6!×((1091)/(12))=65460 7 digit numbers: 7!×((427)/6)=358680 10 digit numbers: 10!×((9577)/(864))=40223400 etc.
$${examples}: \\ $$$$\mathrm{4}\:{digit}\:{numbers}:\:\mathrm{4}!×\mathrm{79}=\mathrm{1896} \\ $$$$\mathrm{5}\:{digit}\:{numbers}:\:\mathrm{5}!×\frac{\mathrm{569}}{\mathrm{6}}=\mathrm{11380} \\ $$$$\mathrm{6}\:{digit}\:{numbers}:\:\mathrm{6}!×\frac{\mathrm{1091}}{\mathrm{12}}=\mathrm{65460} \\ $$$$\mathrm{7}\:{digit}\:{numbers}:\:\mathrm{7}!×\frac{\mathrm{427}}{\mathrm{6}}=\mathrm{358680} \\ $$$$\mathrm{10}\:{digit}\:{numbers}:\:\mathrm{10}!×\frac{\mathrm{9577}}{\mathrm{864}}=\mathrm{40223400} \\ $$$${etc}. \\ $$

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