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Question Number 134960 by bobhans last updated on 09/Mar/21

Number theory 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?

$$\mathrm{Number}\:\mathrm{theory} \\ $$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?

Answered by mr W last updated on 09/Mar/21

(c) k−digit numbers case 1: k=2n XYYY...YY...YYYX X: 1−9 Y: 0−9 ⇒number of numbers=9×10^(n−1) case 2: k=2n+1 XYYY...YZY...YYYX X: 1−9 Y: 0−9 Z: 0−9 ⇒number of numbers=9×10^n generally 9×10^(⌊((k+1)/2)⌋−1) k=2: 9×10^0 =9 numbers k=3: 9×10^1 =90 numbers

$$\left({c}\right)\:{k}−{digit}\:{numbers} \\ $$$${case}\:\mathrm{1}:\:{k}=\mathrm{2}{n} \\ $$$$\mathrm{XYYY}...\mathrm{YY}...\mathrm{YYYX} \\ $$$$\mathrm{X}:\:\mathrm{1}−\mathrm{9} \\ $$$$\mathrm{Y}:\:\mathrm{0}−\mathrm{9} \\ $$$$\Rightarrow{number}\:{of}\:{numbers}=\mathrm{9}×\mathrm{10}^{{n}−\mathrm{1}} \\ $$$${case}\:\mathrm{2}:\:{k}=\mathrm{2}{n}+\mathrm{1} \\ $$$$\mathrm{XYYY}...\mathrm{YZY}...\mathrm{YYYX} \\ $$$$\mathrm{X}:\:\mathrm{1}−\mathrm{9} \\ $$$$\mathrm{Y}:\:\mathrm{0}−\mathrm{9} \\ $$$$\mathrm{Z}:\:\mathrm{0}−\mathrm{9} \\ $$$$\Rightarrow{number}\:{of}\:{numbers}=\mathrm{9}×\mathrm{10}^{{n}} \\ $$$${generally}\:\mathrm{9}×\mathrm{10}^{\lfloor\frac{{k}+\mathrm{1}}{\mathrm{2}}\rfloor−\mathrm{1}} \\ $$$${k}=\mathrm{2}:\:\mathrm{9}×\mathrm{10}^{\mathrm{0}} =\mathrm{9}\:{numbers} \\ $$$${k}=\mathrm{3}:\:\mathrm{9}×\mathrm{10}^{\mathrm{1}} =\mathrm{90}\:{numbers} \\ $$

Commented by Rasheed.Sindhi last updated on 09/Mar/21

Wonderful sir!

$$\mathcal{W}{onderful}\:\:\boldsymbol{{sir}}! \\ $$

Commented by mr W last updated on 09/Mar/21

thanks sir!

$${thanks}\:{sir}! \\ $$

Commented by mr W last updated on 09/Mar/21

it seems you are not so often here as before.

$${it}\:{seems}\:{you}\:{are}\:{not}\:{so}\:{often}\:{here}\:{as} \\ $$$${before}. \\ $$

Commented by Rasheed.Sindhi last updated on 09/Mar/21

Yes sir you′re right. I′m not able to concentrate much.

$${Yes}\:\boldsymbol{{sir}}\:{you}'{re}\:{right}.\:{I}'{m}\:{not}\:{able} \\ $$$${to}\:{concentrate}\:{much}. \\ $$

Commented by bobhans last updated on 09/Mar/21

yess

$$\mathrm{yess} \\ $$

Commented by liberty last updated on 09/Mar/21

how many four digit ones sir? k=2×2 ⇒n=2 the number = 9×10^(2−1) = 90?

$$\mathrm{how}\:\mathrm{many}\:\mathrm{four}\:\mathrm{digit}\:\mathrm{ones}\:\mathrm{sir}? \\ $$$$\mathrm{k}=\mathrm{2}×\mathrm{2}\:\Rightarrow\mathrm{n}=\mathrm{2} \\ $$$$\mathrm{the}\:\mathrm{number}\:=\:\mathrm{9}×\mathrm{10}^{\mathrm{2}−\mathrm{1}} \:=\:\mathrm{90}? \\ $$

Commented by mr W last updated on 10/Mar/21

yes

$${yes} \\ $$

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