Question Number 53902 by mr W last updated on 27/Jan/19
$${How}\:{many}\:{zeros}\:{and}\:{how}\:{many}\:{ones} \\ $$$${are}\:{there}\:{in}\:{the}\:{numbers}\:{from}\:\mathrm{1}\:{to} \\ $$$$\mathrm{9999}? \\ $$$${Example}:\:{in}\:{the}\:{number}\:\mathrm{1010}\:{there} \\ $$$${are}\:\mathrm{2}\:{zeros}\:{and}\:\mathrm{2}\:{ones}. \\ $$
Commented by tanmay.chaudhury50@gmail.com last updated on 27/Jan/19
$${sir}\:{excellent}\:{questiin}...{trying}\:{to}\:{solve}.. \\ $$$${pls}\:{do}\:{not}\:{post}\:{answer}\:{till}\:{the}\:{goal}\:{is}\:{reached} \\ $$$$...{if}\:{not}\:{succeed}...{then}\:{you}\:{pls}\:{upload}\:{answer} \\ $$$$ \\ $$
Commented by Tawa1 last updated on 27/Jan/19
$$\mathrm{Good}\:\mathrm{question} \\ $$
Commented by mr W last updated on 27/Jan/19
$${i}\:{have}\:{not}\:{got}\:{the}\:{answer}\:{yet}.\:{please} \\ $$$${try}\:{to}\:{solve}.\:{thanks}\:{sir}! \\ $$
Answered by tanmay.chaudhury50@gmail.com last updated on 27/Jan/19
![calculation of zdroes... 1)one digit no →use of zero→is 0→0■ 2)Two digit no→when unit place →0 tength digit can be filled by any number→(1,2,3..9) so 9 numbers →9■ 3)Three digit number→unit place→0 tength place can be filled →(1,2,...9)→9ways hundred can be filled →(1,2...8)9 ways so total=1×9×9=81 now use of 0 in this 81 numbers=81■ Three digit no unit place→0 [also tenth place →0 hundred place →can be filled →using( 1,2,3..9) →9 ways →9 numbdrs →e.g[100,200,..900] use of 0 in this 9 numbers=9×2=18■ Three digit no tength place 0→unit plac(1 to9)→ hundred place →any number (1,2,3..9) 9×1×9=81 use of 0→1×81=81■ 4)four dugit no −^× −^× −^× −^0 →9×9×9×1=729 −^× −^× −^0 −^× →9×9×1×9=729 −^× −^0 −^× −^× →9×1×9×9=729 −^× −^× −^0 −^0 →9×9×1×1=81 −^× −^0 −^× −^0 →9×1×9×1=81 −^× −^0 −^0 −^× →9×1×1×9=81 use of 0 [729+729+729]×1■ use of 00[81+81+81]×2■ −^× −^0 −^0 −^0 →9×1×1×1=9 use of 000=9×3=27■ sir pls check till now to get the use of 0 we have to add ■ marked value... pls check.. i have cslculated only use of 0 till now..](Q54003.png)
$${calculation}\:{of}\:{zdroes}... \\ $$$$\left.\mathrm{1}\right){one}\:{digit}\:{no}\:\rightarrow{use}\:{of}\:{zero}\rightarrow{is}\:\mathrm{0}\rightarrow\mathrm{0}\blacksquare \\ $$$$ \\ $$$$\left.\mathrm{2}\right){Two}\:{digit}\:{no}\rightarrow{when}\:{unit}\:{place}\:\rightarrow\mathrm{0} \\ $$$${tength}\:{digit}\:{can}\:{be}\:{filled}\:{by}\:{any}\:{number}\rightarrow\left(\mathrm{1},\mathrm{2},\mathrm{3}..\mathrm{9}\right) \\ $$$${so}\:\mathrm{9}\:{numbers}\:\rightarrow\mathrm{9}\blacksquare \\ $$$$\left.\mathrm{3}\right){Three}\:{digit}\:{number}\rightarrow{unit}\:{place}\rightarrow\mathrm{0} \\ $$$$\:\:\:{tength}\:{place}\:{can}\:{be}\:{filled}\:\rightarrow\left(\mathrm{1},\mathrm{2},...\mathrm{9}\right)\rightarrow\mathrm{9}{ways} \\ $$$${hundred}\:{can}\:{be}\:{filled}\:\rightarrow\left(\mathrm{1},\mathrm{2}...\mathrm{8}\right)\mathrm{9}\:{ways} \\ $$$${so}\:{total}=\mathrm{1}×\mathrm{9}×\mathrm{9}=\mathrm{81} \\ $$$${now}\:{use}\:{of}\:\mathrm{0}\:{in}\:{this}\:\:\mathrm{81}\:{numbers}=\mathrm{81}\blacksquare \\ $$$$ \\ $$$${Three}\:{digit}\:{no}\:{unit}\:{place}\rightarrow\mathrm{0} \\ $$$$\left[{also}\:{tenth}\:{place}\:\rightarrow\mathrm{0}\right. \\ $$$${hundred}\:{place}\:\rightarrow{can}\:{be}\:{filled}\:\rightarrow{using}\left(\:\mathrm{1},\mathrm{2},\mathrm{3}..\mathrm{9}\right) \\ $$$$\rightarrow\mathrm{9}\:{ways}\:\rightarrow\mathrm{9}\:{numbdrs}\:\rightarrow{e}.{g}\left[\mathrm{100},\mathrm{200},..\mathrm{900}\right] \\ $$$${use}\:{of}\:\mathrm{0}\:{in}\:{this}\:\mathrm{9}\:{numbers}=\mathrm{9}×\mathrm{2}=\mathrm{18}\blacksquare \\ $$$$ \\ $$$${Three}\:\boldsymbol{{digit}}\:\boldsymbol{{no}}\:{tength}\:{place}\:\mathrm{0}\rightarrow{unit}\:{plac}\left(\mathrm{1}\:{to}\mathrm{9}\right)\rightarrow \\ $$$${hundred}\:{place}\:\rightarrow{any}\:{number}\:\left(\mathrm{1},\mathrm{2},\mathrm{3}..\mathrm{9}\right) \\ $$$$\mathrm{9}×\mathrm{1}×\mathrm{9}=\mathrm{81} \\ $$$${use}\:{of}\:\mathrm{0}\rightarrow\mathrm{1}×\mathrm{81}=\mathrm{81}\blacksquare \\ $$$$ \\ $$$$\left.\mathrm{4}\right){four}\:{dugit}\:{no} \\ $$$$\overset{×} {−}\:\:\overset{×} {−}\:\overset{×} {−}\:\overset{\mathrm{0}} {−}\rightarrow\mathrm{9}×\mathrm{9}×\mathrm{9}×\mathrm{1}=\mathrm{729} \\ $$$$\overset{×} {−}\:\overset{×} {−}\:\overset{\mathrm{0}} {−}\:\overset{×} {−}\rightarrow\mathrm{9}×\mathrm{9}×\mathrm{1}×\mathrm{9}=\mathrm{729} \\ $$$$\overset{×} {−}\:\overset{\mathrm{0}} {−}\:\overset{×} {−}\:\overset{×} {−}\rightarrow\mathrm{9}×\mathrm{1}×\mathrm{9}×\mathrm{9}=\mathrm{729} \\ $$$$\overset{×} {−}\:\overset{×} {−}\:\overset{\mathrm{0}} {−}\:\overset{\mathrm{0}} {−}\rightarrow\mathrm{9}×\mathrm{9}×\mathrm{1}×\mathrm{1}=\mathrm{81} \\ $$$$\overset{×} {−}\:\overset{\mathrm{0}} {−}\overset{×} {−}\:\overset{\mathrm{0}} {−}\rightarrow\mathrm{9}×\mathrm{1}×\mathrm{9}×\mathrm{1}=\mathrm{81} \\ $$$$\overset{×} {−}\:\overset{\mathrm{0}} {−}\:\overset{\mathrm{0}} {−}\:\overset{×} {−}\rightarrow\mathrm{9}×\mathrm{1}×\mathrm{1}×\mathrm{9}=\mathrm{81} \\ $$$${use}\:{of}\:\mathrm{0}\:\left[\mathrm{729}+\mathrm{729}+\mathrm{729}\right]×\mathrm{1}\blacksquare \\ $$$${use}\:{of}\:\mathrm{00}\left[\mathrm{81}+\mathrm{81}+\mathrm{81}\right]×\mathrm{2}\blacksquare \\ $$$$ \\ $$$$\overset{×} {−}\overset{\mathrm{0}} {−}\:\:\overset{\mathrm{0}} {−}\overset{\mathrm{0}} {−}\:\rightarrow\mathrm{9}×\mathrm{1}×\mathrm{1}×\mathrm{1}=\mathrm{9} \\ $$$${use}\:{of}\:\mathrm{000}=\mathrm{9}×\mathrm{3}=\mathrm{27}\blacksquare \\ $$$$\boldsymbol{{sir}}\:\boldsymbol{{pls}}\:\boldsymbol{{check}}\:\boldsymbol{{till}}\:\boldsymbol{{now}}\:\boldsymbol{{to}}\:\boldsymbol{{get}}\:\boldsymbol{{the}}\:\boldsymbol{{use}}\:\boldsymbol{{of}}\:\mathrm{0} \\ $$$$\boldsymbol{{we}}\:\boldsymbol{{have}}\:\boldsymbol{{to}}\:\boldsymbol{{add}}\:\blacksquare\:{marked}\:{value}... \\ $$$${pls}\:{check}.. \\ $$$$\boldsymbol{{i}}\:\boldsymbol{{have}}\:\boldsymbol{{cslculated}}\:\boldsymbol{{only}}\:\boldsymbol{{use}}\:\boldsymbol{{of}}\:\mathrm{0}\:\boldsymbol{{till}}\:\boldsymbol{{now}}.. \\ $$$$ \\ $$
Commented by mr W last updated on 27/Jan/19
$${correct}\:{sir}! \\ $$
Commented by Otchere Abdullai last updated on 27/Jan/19
$${Brilliant} \\ $$
Commented by tanmay.chaudhury50@gmail.com last updated on 28/Jan/19
$${thank}\:{you}\:{sir}\:{for}\:{your}\:{kind}\:{perusal}\:{through}\:{it}.. \\ $$$${now}\:{i}\:{shall}\:{try}\:{to}\:{solve}\:{for}\:\mathrm{1}... \\ $$
Commented by Tawa1 last updated on 28/Jan/19
$$\mathrm{God}\:\mathrm{bless}\:\mathrm{you}\:\mathrm{sir} \\ $$