Question Number 134960 by bobhans last updated on 09/Mar/21
$$\mathrm{\color{mathbrown}{N}\color{mathbrown}{u}\color{mathbrown}{m}\color{mathbrown}{b}\color{mathbrown}{e}\color{mathbrown}{r}}\color{mathbrown}{\:}\mathrm{\color{mathbrown}{t}\color{mathbrown}{h}\color{mathbrown}{e}\color{mathbrown}{o}\color{mathbrown}{r}\color{mathbrown}{y}} \\ $$ A palindrome is a number that reads the same backwards as forwards, as 3141413. (a)How many two-digit palindromes are there? (b)How many three-digit ones? (c)How many k-digits ones?\\n
Answered by mr W last updated on 09/Mar/21
$$\left({c}\right)\:{k}−{digit}\:{numbers} \\ $$ $${case}\:\mathrm{1}:\:{k}=\mathrm{2}{n} \\ $$ $$\mathrm{XYYY}...\mathrm{Y\color{mathblue}{Y}}\color{mathblue}{.}\color{mathblue}{.}\color{mathblue}{.}\mathrm{\color{mathblue}{Y}\color{mathblue}{Y}\color{mathblue}{Y}\color{mathblue}{X}} \\ $$ $$\mathrm{X}:\:\mathrm{1}−\mathrm{9} \\ $$ $$\mathrm{Y}:\:\mathrm{0}−\mathrm{9} \\ $$ $$\Rightarrow{number}\:{of}\:{numbers}=\mathrm{9}×\mathrm{10}^{{n}−\mathrm{1}} \\ $$ $${case}\:\mathrm{2}:\:{k}=\mathrm{2}{n}+\mathrm{1} \\ $$ $$\mathrm{XYYY}...\mathrm{Y\color{mathbrown}{Z}\color{mathblue}{Y}}\color{mathblue}{.}\color{mathblue}{.}\color{mathblue}{.}\mathrm{\color{mathblue}{Y}\color{mathblue}{Y}\color{mathblue}{Y}\color{mathblue}{X}} \\ $$ $$\mathrm{X}:\:\mathrm{1}−\mathrm{9} \\ $$ $$\mathrm{Y}:\:\mathrm{0}−\mathrm{9} \\ $$ $$\mathrm{Z}:\:\mathrm{0}−\mathrm{9} \\ $$ $$\Rightarrow{number}\:{of}\:{numbers}=\mathrm{9}×\mathrm{10}^{{n}} \\ $$ $${generally}\:\mathrm{\color{mathred}{9}}\color{mathred}{×}\mathrm{\color{mathred}{1}\color{mathred}{0}}^{\color{mathred}{\lfloor}\frac{{\color{mathred}{k}}\color{mathred}{+}\mathrm{\color{mathred}{1}}}{\mathrm{\color{mathred}{2}}}\color{mathred}{\rfloor}\color{mathred}{−}\mathrm{\color{mathred}{1}}} \\ $$ $${k}=\mathrm{2}:\:\mathrm{9}×\mathrm{10}^{\mathrm{0}} =\mathrm{9}\:{numbers} \\ $$ $${k}=\mathrm{3}:\:\mathrm{9}×\mathrm{10}^{\mathrm{1}} =\mathrm{90}\:{numbers} \\ $$
Commented byRasheed.Sindhi last updated on 09/Mar/21
$$\mathcal{\color{mathred}{W}}{onderful}\:\:\boldsymbol{{sir}}! \\ $$
Commented bymr W last updated on 09/Mar/21
$${thanks}\:{sir}! \\ $$
Commented bymr W last updated on 09/Mar/21
$${it}\:{seems}\:{you}\:{are}\:{not}\:{so}\:{often}\:{here}\:{as} \\ $$ $${before}. \\ $$
Commented byRasheed.Sindhi last updated on 09/Mar/21
$${Yes}\:\boldsymbol{{sir}}\:{you}'{re}\:{right}.\:{I}'{m}\:{not}\:{able} \\ $$ $${to}\:{concentrate}\:{much}. \\ $$
Commented bybobhans last updated on 09/Mar/21
$$\mathrm{yess} \\ $$
Commented byliberty last updated on 09/Mar/21
$$\mathrm{how}\:\mathrm{many}\:\mathrm{four}\:\mathrm{digit}\:\mathrm{ones}\:\mathrm{sir}? \\ $$ $$\mathrm{k}=\mathrm{2}×\mathrm{2}\:\Rightarrow\mathrm{n}=\mathrm{2} \\ $$ $$\mathrm{the}\:\mathrm{number}\:=\:\mathrm{9}×\mathrm{10}^{\mathrm{2}−\mathrm{1}} \:=\:\mathrm{90}? \\ $$
Commented bymr W last updated on 10/Mar/21
$${yes} \\ $$