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Question Number 198576 by mr W last updated on 22/Oct/23
How many numbers with a maximum of 5 digits, greater than 4000, can be formed with the digits 2, 3, 4, 5, 6; if repetition is allowed for 2 and 3 only?
Commented by mr W last updated on 22/Oct/23
$${Q}\mathrm{198242}\:{reposted}\:{for}\:{alternative} \\ $$$${solutions} \\ $$
Commented by MM42 last updated on 22/Oct/23
$${Sir}\:{W} \\ $$$${you}\:{are}\:{right}.{i}\:{was}\:{not}\:{careful}. \\ $$$${thank}\:{you}\:{for}\:{your}\:{consideration}. \\ $$
Commented by mr W last updated on 22/Oct/23
$${you}\:{are}\:{welcome}\:{sir}! \\ $$
Commented by nikif99 last updated on 23/Oct/23
$$ \\ $$$${A}\:{more}\:{complicated}\:{solution} \\ $$$$ \\ $$$$\boldsymbol{{I}}.\:\boldsymbol{{four}}\:\boldsymbol{{digit}}\:\boldsymbol{{numbers}} \\ $$$$\begin{matrix}{\mathrm{1}{st}}&{\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\mathrm{2}{nd}}&{\:\:\:\:\:\:\:\:\mathrm{3}{rd}}&{\:\mathrm{4}{th}}&{\:\:\:\:\:\:{cases}^{\bigstar} }\end{matrix} \\ $$$$\mathrm{4}{or}\mathrm{5}{or}\mathrm{6}^{\bigstar} \begin{cases}{\mathrm{2}{or}\mathrm{3}\:\:\:\begin{cases}{\mathrm{2}{or}\mathrm{3\begin{cases}{\mathrm{2}{or}\mathrm{3}\:\:\:\:\:\:\:\left(\mathrm{3}×\mathrm{2}×\mathrm{2}×\mathrm{2}=\mathrm{24}\right)}\\{\mathrm{5}{or}\mathrm{6}\:\:\:\:\:\:\:\left(\mathrm{24}\right)}\end{cases}}}\\{\mathrm{5}{or}\mathrm{6\begin{cases}{\mathrm{2}{or}\mathrm{3}\:\:\:\:\:\:\:\left(\mathrm{24}\right)}\\{\mathrm{6}\:\:\:\:\:\:\:\:\:\:\:\:\:\:\left(\mathrm{12}\right)}\end{cases}}}\end{cases}}\\{\mathrm{5}{or}\mathrm{6}^{\bigstar} \begin{cases}{\mathrm{2}{or}\mathrm{3\begin{cases}{\mathrm{2}{or}\mathrm{3}\:\:\:\:\:\:\:\left(\mathrm{24}\right)}\\{\mathrm{6}^{\bigstar} \:\:\:\:\:\:\:\:\:\:\:\left(\mathrm{12}\right)}\end{cases}}}\\{\mathrm{6}\:\:\:\:\:\:\:\:\begin{cases}{\mathrm{2}{or}\mathrm{3}\:\:\:\:\:\:\left(\mathrm{12}\right)}\\{\:\:×\:\:\:\:\:\:\:\:\:\:\left(\mathrm{0}\right)}\end{cases}}\end{cases}}\end{cases} \\ $$$$\bigstar:\:{the}\:{product}\:{of}\:{all}\:{possible}\:{cases}\:{through}\:{a}\:{routing}\:{of}\:{digits}. \\ $$$${Subtotal}\:\mathrm{132} \\ $$$$ \\ $$$$\boldsymbol{{II}}.\:\boldsymbol{{five}}\:\boldsymbol{{digit}}\:\boldsymbol{{numbers}} \\ $$$$\begin{matrix}{\mathrm{1}{st}}&{\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\mathrm{2}{nd}}&{\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\mathrm{3}{rd}}&{\:\:\:\:\:\:\:\:\:\:\:\:\:\mathrm{4}{th}}&{\:\:\:\:\:\:\:\:\:\mathrm{5}{th}}&{\:\:\:\:{cases}}\end{matrix} \\ $$$$\begin{matrix}{\mathrm{2}{or}\mathrm{3\begin{cases}{\mathrm{2}{or}\mathrm{3}\:\:\:\:\:\:\:\:\:\:\:\begin{cases}{\mathrm{2}{or}\mathrm{3}\:\:\:\:\:\:\:\begin{cases}{\mathrm{2}{or}\mathrm{3}\:\:\:\:\:\:\:\begin{cases}{\mathrm{2}{or}\mathrm{3}\:\:\:\:\:\:\:\:\:\left(\mathrm{32}\right)}\\{\mathrm{4}{or}\mathrm{5}{or}\mathrm{6}\:\:\left(\mathrm{48}\right)}\end{cases}}\\{\mathrm{4}{or}\mathrm{5}{or}\mathrm{6\begin{cases}{\mathrm{2}{or}\mathrm{3}\:\:\:\:\:\:\:\:\:\left(\mathrm{48}\right)}\\{\mathrm{5}{or}\mathrm{6}\:\:\:\:\:\:\:\:\:\left(\mathrm{48}\right)}\end{cases}}}\end{cases}}\\{\mathrm{4}{or}\mathrm{5}{or}\mathrm{6\begin{cases}{\mathrm{2}{or}\mathrm{3}\:\:\:\:\:\:\:\begin{cases}{\mathrm{2}{or}\mathrm{3}\:\:\:\:\:\:\:\:\:\:\left(\mathrm{48}\right)}\\{\mathrm{5}{or}\mathrm{6}\:\:\:\:\:\:\:\:\:\:\left(\mathrm{48}\right)}\end{cases}}\\{\mathrm{5}{or}\mathrm{6}\:\:\:\:\:\:\:\begin{cases}{\mathrm{2}{or}\mathrm{3}\:\:\:\:\:\:\:\:\:\:\:\left(\mathrm{48}\right)}\\{\mathrm{6}\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\left(\mathrm{24}\right)}\end{cases}}\end{cases}}}\end{cases}}\\{\mathrm{4}{or}\mathrm{5}{or}\mathrm{6}^{\bigstar} \begin{cases}{\mathrm{2}{or}\mathrm{3}\:\:\:\:\:\:\:\:\begin{cases}{\mathrm{2}{or}\mathrm{3}\:\:\:\:\:\:\begin{cases}{\mathrm{2}{or}\mathrm{3}\:\:\:\:\:\:\:\:\:\:\:\left(\mathrm{48}\right)}\\{\mathrm{5}{or}\mathrm{6}\:\:\:\:\:\:\:\:\:\:\:\left(\mathrm{48}\right)}\end{cases}}\\{\mathrm{5}{or}\mathrm{6}\:\:\:\:\:\:\begin{cases}{\mathrm{2}{or}\mathrm{3}\:\:\:\:\:\:\:\:\:\:\:\left(\mathrm{48}\right)}\\{\mathrm{6}\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\left(\mathrm{24}\right)}\end{cases}}\end{cases}}\\{\mathrm{5}{or}\mathrm{6}^{\bigstar} \:\:\:\:\:\begin{cases}{\mathrm{2}{or}\mathrm{3}\:\:\:\:\:\:\begin{cases}{\mathrm{2}{or}\mathrm{3}\:\:\:\:\:\:\:\:\:\:\left(\mathrm{48}\right)}\\{\mathrm{6}^{\bigstar} \:\:\:\:\:\:\:\:\:\:\:\:\:\:\left(\mathrm{24}\right)}\end{cases}}\\{\mathrm{6}\:\:\:\:\:\:\:\:\:\:\:\:\:\:\begin{cases}{\mathrm{2}\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\left(\mathrm{12}\right)}\\{\mathrm{3}\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\left(\mathrm{12}\right)}\end{cases}}\end{cases}}\end{cases}}\end{cases}}}\end{matrix} \\ $$$${Subtotal}\:\mathrm{608} \\ $$$$\mathrm{4}{or}\mathrm{5}{or}\mathrm{6}^{\bigstar} \begin{cases}{\mathrm{2}{or}\mathrm{3}\:\:\:\:\:\:\begin{cases}{\mathrm{2}{or}\mathrm{3\begin{cases}{\mathrm{2}{or}\mathrm{3}\:\:\:\:\:\:\:\:\:\:\:\:\begin{cases}{\mathrm{2}{or}\mathrm{3}\:\:\:\:\:\:\:\left(\mathrm{48}\right)}\\{\mathrm{5}{or}\mathrm{6}\:\:\:\:\:\:\:\left(\mathrm{48}\right)}\end{cases}}\\{\mathrm{5}{or}\mathrm{6}\:\:\:\:\:\:\:\:\:\:\:\:\begin{cases}{\mathrm{2}{or}\mathrm{3}\:\:\:\:\:\:\:\left(\mathrm{48}\right)}\\{\mathrm{6}\:\:\:\:\:\:\:\:\:\:\:\:\:\:\left(\mathrm{24}\right)}\end{cases}}\end{cases}}}\\{\mathrm{5}{or}\mathrm{6\begin{cases}{\mathrm{2}{or}\mathrm{3}\:\:\:\:\:\:\:\:\:\:\:\:\begin{cases}{\mathrm{2}{or}\mathrm{3}\:\:\:\:\:\:\:\left(\mathrm{48}\right)}\\{\mathrm{6}\:\:\:\:\:\:\:\:\:\:\:\:\:\:\left(\mathrm{24}\right)}\end{cases}}\\{\mathrm{6}\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\begin{cases}{\mathrm{2}\:\:\:\:\:\:\:\:\:\:\:\:\:\left(\mathrm{12}\right)}\\{\mathrm{3}\:\:\:\:\:\:\:\:\:\:\:\:\:\left(\mathrm{12}\right)}\end{cases}}\end{cases}}}\end{cases}}\\{\mathrm{5}{or}\mathrm{6}^{\bigstar} \:\:\:\:\:\:\begin{cases}{\mathrm{2}{or}\mathrm{3\begin{cases}{\mathrm{2}{or}\mathrm{3}\:\:\:\:\:\:\:\:\:\:\:\:\begin{cases}{\mathrm{2}{or}\mathrm{3}\:\:\:\:\:\:\left(\mathrm{48}\right)}\\{\mathrm{6}\:\:\:\:\:\:\:\:\:\:\:\:\:\left(\mathrm{24}\right)}\end{cases}}\\{\mathrm{6}^{\bigstar} \:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\begin{cases}{\mathrm{2}\:\:\:\:\:\:\:\:\:\:\:\:\:\:\left(\mathrm{8}\right)}\\{\mathrm{3}\:\:\:\:\:\:\:\:\:\:\:\:\:\:\left(\mathrm{8}\right)}\end{cases}}\end{cases}}}\\{\mathrm{6}\:\:\:\:\:\:\:\begin{cases}{\mathrm{2}\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\begin{cases}{\mathrm{2}\:\:\:\:\:\:\:\:\:\:\:\:\:\:\left(\mathrm{8}\right)}\\{\mathrm{3}\:\:\:\:\:\:\:\:\:\:\:\:\:\:\left(\mathrm{8}\right)}\end{cases}}\\{\mathrm{3}\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\begin{cases}{\mathrm{2}\:\:\:\:\:\:\:\:\:\:\:\:\:\:\left(\mathrm{8}\right)}\\{\mathrm{3}\:\:\:\:\:\:\:\:\:\:\:\:\:\:\left(\mathrm{8}\right)}\end{cases}}\end{cases}}\end{cases}}\end{cases} \\ $$$${Subtotal}\:\mathrm{384} \\ $$$${Total}\:\mathrm{1124} \\ $$$$ \\ $$$$\bigstar:\:{if}\:{for}\:{exemple}\:\mathrm{4}\:{is}\:{selected}\:{initialy},\:{then}\:{one}\:{of}\:\left\{\mathrm{5},\mathrm{6}\right\}\:{is}\:{selected}\:{next}, \\ $$$${and}\:{finally}\:{the}\:{last}\:{one}. \\ $$
Commented by mr W last updated on 23/Oct/23
$${method}\:{using}\:{generating}\:{function} \\ $$$${seems}\:{to}\:{be}\:{camplicated}\:{at}\:{first} \\ $$$${glance},\:{but}\:{actually}\:{this}\:{solution} \\ $$$${method}\:\:{is}\:{general}\:{and}\:{thus}\:{very} \\ $$$${powerful}.\:{try}\:{to}\:{find}\:{the}\:{number} \\ $$$${of}\:\mathrm{7}−{digit}\:{numbers}\:{with}\:{exactly} \\ $$$${the}\:{same}\:{conditions}.\:{can}\:{you}\:{give} \\ $$$${a}\:{solution}\:{for}\:{this}\:{case}? \\ $$
Commented by mr W last updated on 23/Oct/23
Commented by nikif99 last updated on 23/Oct/23
$${It}\:{is}\:{quite}\:{difficult}\:{to}\:{manage}\:{manually} \\ $$$${such}\:{a}\:{tree}\:{with}\:\mathrm{7}\:{levels}. \\ $$$${Trying}\:{another}\:{approach},\:{I}\:{used}\: \\ $$$${self}−{developped}\:{software}\:{via}\:{Fortran} \\ $$$${and}\:{I}\:{found}\:\mathrm{8}.\mathrm{864}\:{cases}\:{for}\:\mathrm{7}−{digit}\: \\ $$$${numbers}\:\left({and}\:\mathrm{3}.\mathrm{040}\:{for}\:\mathrm{6}−{digit}\right). \\ $$$${P}.{S}.\:{I}\:{do}\:{not}\:{know}\:{what}\:{other}\:{members} \\ $$$${see}\:{on}\:{my}\:{solution}\:{comment};\: \\ $$$${I}\:{personally}\:{see}\:{only}\:{the}\:{title}\:{in}\:{bold} \\ $$$$\left(``\mathrm{4}−{digit}\:{numbers}''\right)\:{without}\:{analysis} \\ $$$${with}\:{opening}\:{curly}\:{braces}. \\ $$
Commented by mr W last updated on 23/Oct/23
$$\mathrm{8864}\:{numbers}\:{with}\:\mathrm{7}\:{digits},\:{this}\:{is} \\ $$$${correct},\:{thanks}\:{alot}!\:{i}\:{also}\:{got}\:{this} \\ $$$${result},\:{very}\:{easily}\:{with}\:{my}\:{method} \\ $$$${applying}\:{generating}\:{function}.\:{the} \\ $$$${result}\:{is}\:{the}\:{coefficient}\:{of}\:{x}^{\mathrm{7}} \:{term} \\ $$$${in}\:\mathrm{7}!{e}^{\mathrm{2}{x}} \left(\mathrm{1}+{x}\right)^{\mathrm{3}} ,\:{which}\:{is}\:\mathrm{8864}. \\ $$$${similarly}\:{for}\:\mathrm{6}−{digit}\:{numbers}\:{it}\:{is} \\ $$$${the}\:{coefficient}\:{of}\:{x}^{\mathrm{6}} \:{term}\:{in}\: \\ $$$$\mathrm{6}!{e}^{\mathrm{2}{x}} \left(\mathrm{1}+{x}\right)^{\mathrm{3}} ,\:{which}\:{is}\:\mathrm{3040}. \\ $$
Commented by nikif99 last updated on 23/Oct/23
Commented by mr W last updated on 23/Oct/23
$${nice}\:{prepared}! \\ $$
Answered by mr W last updated on 23/Oct/23
$$\mathrm{2}\:{and}\:\mathrm{3}\:{can}\:{be}\:{used}\:{zero}\:{time},\:{one} \\ $$$${time}\:{or}\:{any}\:{times}\:\Rightarrow\left(\mathrm{1}+{x}+{x}^{\mathrm{2}} +{x}^{\mathrm{3}} +...\right) \\ $$$$\mathrm{4},\:\mathrm{5}\:{and}\:\mathrm{6}\:{can}\:{only}\:{be}\:{used}\:{zero}\:{time}\: \\ $$$${or}\:{one}\:{time}\:\Rightarrow\left(\mathrm{1}+{x}\right) \\ $$$$ \\ $$$$\underline{\mathrm{5}−\boldsymbol{{digit}}\:\boldsymbol{{numbers}}:} \\ $$$${number}\:{of}\:{such}\:{numbers}\:{is}\:{the} \\ $$$${coefficient}\:{of}\:{x}^{\mathrm{5}} \:{term}\:{in} \\ $$$$\mathrm{5}!\left(\mathrm{1}+{x}+\frac{{x}^{\mathrm{2}} }{\mathrm{2}!}+\frac{{x}^{\mathrm{3}} }{\mathrm{3}!}+...\right)^{\mathrm{2}} \left(\mathrm{1}+{x}\right)^{\mathrm{3}} \\ $$$$=\mathrm{5}!{e}^{\mathrm{2}{x}} \left(\mathrm{1}+{x}\right)^{\mathrm{3}} \\ $$$${which}\:{is}\:\mathrm{992}.\: \\ $$$${that}\:{means}\:{we}\:{can}\:{form}\:\mathrm{992}\:{such}\: \\ $$$$\mathrm{5}−{digit}\:{numbers}. \\ $$
Commented by Tawa11 last updated on 23/Oct/23
$$\mathrm{Thanks}\:\mathrm{sir}. \\ $$$$\mathrm{I}\:\mathrm{appreciate}\:\mathrm{your}\:\mathrm{time},\:\mathrm{God}\:\mathrm{bless}\:\mathrm{you}\:\mathrm{sir}. \\ $$
Commented by mr W last updated on 22/Oct/23
Commented by mr W last updated on 22/Oct/23
$$\underline{\mathrm{4}−\boldsymbol{{digit}}\:\boldsymbol{{numbers}}:} \\ $$$${the}\:{first}\:{digit}\:{must}\:{be}\:\mathrm{4},\:\mathrm{5}\:{or}\:\mathrm{6}. \\ $$$${we}\:{select}\:{at}\:{first}\:{the}\:{first}\:{digit}, \\ $$$${there}\:{are}\:\mathrm{3}\:{ways}.\:{then}\:{we}\:{form}\: \\ $$$$\mathrm{3}−{digit}\:{numbers}\:{with}\:{the}\:{other}\:\mathrm{4} \\ $$$${digits}.\:{similarly}\:{to}\:{above},\:{the}\:{number} \\ $$$${of}\:{such}\:{numbers}\:{is}\:{the}\:{coefficient} \\ $$$${of}\:{x}^{\mathrm{3}} \:{term}\:{in}\:\mathrm{3}!{e}^{\mathrm{2}{x}} \left(\mathrm{1}+{x}\right)^{\mathrm{2}} ,\:{which}\:{is} \\ $$$$\mathrm{44}.\:{so}\:{totally}\:{we}\:{can}\:{form}\: \\ $$$$\mathrm{3}×\mathrm{44}=\mathrm{132}\:{valid}\:\mathrm{4}−{digit}\:{numbers}. \\ $$
Commented by mr W last updated on 22/Oct/23
Commented by mr W last updated on 22/Oct/23
$${all}\:{together}:\:\mathrm{992}+\mathrm{132}=\mathrm{1124}\:\checkmark \\ $$