Question and Answers Forum

All Questions      Topic List

Permutation and Combination Questions

Previous in All Question      Next in All Question      

Previous in Permutation and Combination      Next in Permutation and Combination      

Question Number 105791 by mr W last updated on 01/Aug/20

From 1 to 12345, how many numbers contain the digit 0? Find the number of zeros in all these numbers. Example: 10020 has three zeros.

$${From}\:\mathrm{1}\:{to}\:\mathrm{12345},\:{how}\:{many}\:{numbers} \\ $$$${contain}\:{the}\:{digit}\:\mathrm{0}?\:{Find}\:{the}\:{number} \\ $$$${of}\:{zeros}\:{in}\:{all}\:{these}\:{numbers}. \\ $$$${Example}:\:\mathrm{10020}\:{has}\:{three}\:{zeros}. \\ $$

Commented by PRITHWISH SEN 2 last updated on 01/Aug/20

189+2700+1630 = 4519 The answer is 4519 please check

$$\mathrm{189}+\mathrm{2700}+\mathrm{1630}\:=\:\mathrm{4519} \\ $$$$\mathrm{The}\:\mathrm{answer}\:\mathrm{is}\:\mathrm{4519}\:\:\boldsymbol{\mathrm{please}}\:\boldsymbol{\mathrm{check}} \\ $$

Answered by mr W last updated on 02/Aug/20

N=number of numbers containing at least one zero S=number of zeros in the numbers X,Y,Z=placeholder non−zero digits one digit numbers: X ⇒N=0, S=0 two digit numbers: X0 ⇒N=9, S=9 three digit numbers: X0Y, XY0 ⇒N=2×9^2 =162, S=162 X00 ⇒N=9, S=2×9=18 ⇒total N=162+9=171 ⇒total S=162+18=180 four digit numbers: X0YZ,XY0Z,XYZ0 ⇒N=3×9^3 =2187, S=2187 X00Y,XY00,X0Y0 ⇒N=3×9^2 =243, S=486 X000 ⇒N=9, S=27 ⇒total N=2187+243+9=2439 ⇒total S=2187+486+27=2700 five digit numbers: 10000 ⇒N=1, S=4 1X000,10X00,100X0,1000X ⇒N=2+9+9+9=29, S=87 1XY00,10XY0,100XY,1X0Y0,1X00Y,10X0Y ⇒N=(9+3)+3×9^2 +2×2×9=291 ⇒S=2×291=582 10XYZ,1X0YZ,1XY0Z,1XYZ0 ⇒N=9^3 +2×9^2 +(9^2 +3×9)+(9^2 +2×9+4)=1102 ⇒S=1102 ⇒total N=1+29+291+1102=1423 ⇒total S=4+87+582+1102=1775 totally: N=9+171+2439+1423=4042 S=9+180+2700+1775=4664

$${N}={number}\:{of}\:{numbers}\:{containing} \\ $$$${at}\:{least}\:{one}\:{zero} \\ $$$${S}={number}\:{of}\:{zeros}\:{in}\:{the}\:{numbers} \\ $$$$ \\ $$$${X},{Y},{Z}={placeholder}\:{non}−{zero}\:{digits} \\ $$$$ \\ $$$$\boldsymbol{{one}}\:\boldsymbol{{digit}}\:\boldsymbol{{numbers}}: \\ $$$${X}\:\Rightarrow{N}=\mathrm{0},\:{S}=\mathrm{0} \\ $$$$ \\ $$$$\boldsymbol{{two}}\:\boldsymbol{{digit}}\:\boldsymbol{{numbers}}: \\ $$$${X}\mathrm{0}\:\Rightarrow{N}=\mathrm{9},\:{S}=\mathrm{9} \\ $$$$ \\ $$$$\boldsymbol{{three}}\:\boldsymbol{{digit}}\:\boldsymbol{{numbers}}: \\ $$$${X}\mathrm{0}{Y},\:{XY}\mathrm{0}\:\Rightarrow{N}=\mathrm{2}×\mathrm{9}^{\mathrm{2}} =\mathrm{162},\:{S}=\mathrm{162} \\ $$$${X}\mathrm{00}\:\Rightarrow{N}=\mathrm{9},\:{S}=\mathrm{2}×\mathrm{9}=\mathrm{18} \\ $$$$\Rightarrow{total}\:{N}=\mathrm{162}+\mathrm{9}=\mathrm{171} \\ $$$$\Rightarrow{total}\:{S}=\mathrm{162}+\mathrm{18}=\mathrm{180} \\ $$$$ \\ $$$$\boldsymbol{{four}}\:\boldsymbol{{digit}}\:\boldsymbol{{numbers}}: \\ $$$${X}\mathrm{0}{YZ},{XY}\mathrm{0}{Z},{XYZ}\mathrm{0}\:\Rightarrow{N}=\mathrm{3}×\mathrm{9}^{\mathrm{3}} =\mathrm{2187},\:{S}=\mathrm{2187} \\ $$$${X}\mathrm{00}{Y},{XY}\mathrm{00},{X}\mathrm{0}{Y}\mathrm{0}\:\Rightarrow{N}=\mathrm{3}×\mathrm{9}^{\mathrm{2}} =\mathrm{243},\:{S}=\mathrm{486} \\ $$$${X}\mathrm{000}\:\Rightarrow{N}=\mathrm{9},\:{S}=\mathrm{27} \\ $$$$\Rightarrow{total}\:{N}=\mathrm{2187}+\mathrm{243}+\mathrm{9}=\mathrm{2439} \\ $$$$\Rightarrow{total}\:{S}=\mathrm{2187}+\mathrm{486}+\mathrm{27}=\mathrm{2700} \\ $$$$ \\ $$$$\boldsymbol{{five}}\:\boldsymbol{{digit}}\:\boldsymbol{{numbers}}: \\ $$$$\mathrm{10000}\:\Rightarrow{N}=\mathrm{1},\:{S}=\mathrm{4} \\ $$$$\mathrm{1}{X}\mathrm{000},\mathrm{10}{X}\mathrm{00},\mathrm{100}{X}\mathrm{0},\mathrm{1000}{X} \\ $$$$\Rightarrow{N}=\mathrm{2}+\mathrm{9}+\mathrm{9}+\mathrm{9}=\mathrm{29},\:{S}=\mathrm{87} \\ $$$$\mathrm{1}{XY}\mathrm{00},\mathrm{10}{XY}\mathrm{0},\mathrm{100}{XY},\mathrm{1}{X}\mathrm{0}{Y}\mathrm{0},\mathrm{1}{X}\mathrm{00}{Y},\mathrm{10}{X}\mathrm{0}{Y} \\ $$$$\Rightarrow{N}=\left(\mathrm{9}+\mathrm{3}\right)+\mathrm{3}×\mathrm{9}^{\mathrm{2}} +\mathrm{2}×\mathrm{2}×\mathrm{9}=\mathrm{291} \\ $$$$\Rightarrow{S}=\mathrm{2}×\mathrm{291}=\mathrm{582} \\ $$$$\mathrm{10}{XYZ},\mathrm{1}{X}\mathrm{0}{YZ},\mathrm{1}{XY}\mathrm{0}{Z},\mathrm{1}{XYZ}\mathrm{0} \\ $$$$\Rightarrow{N}=\mathrm{9}^{\mathrm{3}} +\mathrm{2}×\mathrm{9}^{\mathrm{2}} +\left(\mathrm{9}^{\mathrm{2}} +\mathrm{3}×\mathrm{9}\right)+\left(\mathrm{9}^{\mathrm{2}} +\mathrm{2}×\mathrm{9}+\mathrm{4}\right)=\mathrm{1102} \\ $$$$\Rightarrow{S}=\mathrm{1102} \\ $$$$\Rightarrow{total}\:{N}=\mathrm{1}+\mathrm{29}+\mathrm{291}+\mathrm{1102}=\mathrm{1423} \\ $$$$\Rightarrow{total}\:{S}=\mathrm{4}+\mathrm{87}+\mathrm{582}+\mathrm{1102}=\mathrm{1775} \\ $$$$ \\ $$$$\boldsymbol{{totally}}: \\ $$$${N}=\mathrm{9}+\mathrm{171}+\mathrm{2439}+\mathrm{1423}=\mathrm{4042} \\ $$$${S}=\mathrm{9}+\mathrm{180}+\mathrm{2700}+\mathrm{1775}=\mathrm{4664} \\ $$

Commented by PRITHWISH SEN 2 last updated on 01/Aug/20

considering 2 zero of 5 digits the no. of zero will be = 285×2=570 instead of 285×3= 855 ∴ the total zero = 4519 please check sir .

$$\mathrm{considering}\:\mathrm{2}\:\mathrm{zero}\:\mathrm{of}\:\mathrm{5}\:\mathrm{digits} \\ $$$$\mathrm{the}\:\mathrm{no}.\:\mathrm{of}\:\mathrm{zero}\:\mathrm{will}\:\mathrm{be}\:=\:\mathrm{285}×\mathrm{2}=\mathrm{570}\:\boldsymbol{\mathrm{instead}}\:\boldsymbol{\mathrm{of}} \\ $$$$\mathrm{285}×\mathrm{3}=\:\mathrm{855}\:\:\therefore\:\boldsymbol{\mathrm{the}}\:\boldsymbol{\mathrm{total}}\:\boldsymbol{\mathrm{zero}}\:=\:\mathrm{4519} \\ $$$$\:\boldsymbol{\mathrm{please}}\:\boldsymbol{\mathrm{check}}\:\boldsymbol{\mathrm{sir}}\:. \\ $$$$ \\ $$

Commented by mr W last updated on 01/Aug/20

it was wrong for five digit numbers. now the answer is correct.

$${it}\:{was}\:{wrong}\:{for}\:{five}\:{digit}\:{numbers}. \\ $$$${now}\:{the}\:{answer}\:{is}\:{correct}. \\ $$

Terms of Service

Privacy Policy

Contact: info@tinkutara.com