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

$$\mathrm{N}\mathrm{u}\mathrm{m}\mathrm{b}\mathrm{e}\mathrm{r}\mathrm{t}\mathrm{h}\mathrm{e}\mathrm{o}\mathrm{r}\mathrm{y}$$ 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

$$\left(c\right)k-digitnumbers$$ $$case1:k=2n$$ $$\mathrm{XYYY}...\mathrm{Y}\mathrm{Y}...\mathrm{Y}\mathrm{Y}\mathrm{Y}\mathrm{X}$$ $$\mathrm{X}:1-9$$ $$\mathrm{Y}:0-9$$ $$\Rightarrow numberofnumbers=9\times {10}^{n-1}$$ $$case2:k=2n+1$$ $$\mathrm{XYYY}...\mathrm{Y}\mathrm{Z}\mathrm{Y}...\mathrm{Y}\mathrm{Y}\mathrm{Y}\mathrm{X}$$ $$\mathrm{X}:1-9$$ $$\mathrm{Y}:0-9$$ $$\mathrm{Z}:0-9$$ $$\Rightarrow numberofnumbers=9\times {10}^n$$ $$generally9\times {10}^{\lfloor \frac{k+1}{2}\rfloor -1}$$ $$k=2:9\times {10}^0=9numbers$$ $$k=3:9\times {10}^1=90numbers$$

Commented byRasheed.Sindhi last updated on 09/Mar/21

$$\mathcal{W}onderful\mathit{s}\mathit{i}\mathit{r}!$$

Commented bymr W last updated on 09/Mar/21

$$thankssir!$$

Commented bymr W last updated on 09/Mar/21

$$itseemsyouarenotsooftenhereas$$ $$before.$$

Commented byRasheed.Sindhi last updated on 09/Mar/21

$$Yes\mathit{s}\mathit{i}\mathit{r}{you}^{\prime }reright.I^{\prime }mnotable$$ $$toconcentratemuch.$$

Commented bybobhans last updated on 09/Mar/21

$$\mathrm{yess}$$

Commented byliberty last updated on 09/Mar/21

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

Commented bymr W last updated on 10/Mar/21

$$yes$$

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