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Question Number 48543 by Pk1167156@gmail.com last updated on 25/Nov/18

The number of times the digit 3 will be written when listing the integers from 1 to 1000 is

$$\mathrm{The}\:\mathrm{number}\:\mathrm{of}\:\mathrm{times}\:\mathrm{the}\:\mathrm{digit}\:\mathrm{3}\:\mathrm{will} \\ $$$$\mathrm{be}\:\mathrm{written}\:\mathrm{when}\:\mathrm{listing}\:\mathrm{the}\:\mathrm{integers} \\ $$$$\mathrm{from}\:\mathrm{1}\:\mathrm{to}\:\mathrm{1000}\:\mathrm{is} \\ $$

Answered by mr W last updated on 25/Nov/18

one−digit−numbers: 3⇒1 number⇒1 time two−digit−numbers: x3⇒ 8 numbers⇒8 times 3x⇒ 9 numbers⇒9 times 33⇒ 1 number⇒2 times three−digit−numbers: xy3 ⇒8×9=72 numbers⇒72 times x3y ⇒8×9=72 numbers⇒72 times 3xy ⇒9×9=81 numbers⇒81 times x33 ⇒8 numbers⇒16 times 33x ⇒9 numbers⇒18 times 3x3 ⇒9 numbers⇒18 times 333 ⇒1 numbers⇒3 times Σtimes=1+8+9+2+72+72+81+16+18+18+3 Σtimes=300

$${one}−{digit}−{numbers}: \\ $$$$\mathrm{3}\Rightarrow\mathrm{1}\:{number}\Rightarrow\mathrm{1}\:{time} \\ $$$$ \\ $$$${two}−{digit}−{numbers}: \\ $$$${x}\mathrm{3}\Rightarrow\:\mathrm{8}\:{numbers}\Rightarrow\mathrm{8}\:{times} \\ $$$$\mathrm{3}{x}\Rightarrow\:\mathrm{9}\:{numbers}\Rightarrow\mathrm{9}\:{times} \\ $$$$\mathrm{33}\Rightarrow\:\mathrm{1}\:{number}\Rightarrow\mathrm{2}\:{times} \\ $$$$ \\ $$$${three}−{digit}−{numbers}: \\ $$$${xy}\mathrm{3}\:\Rightarrow\mathrm{8}×\mathrm{9}=\mathrm{72}\:{numbers}\Rightarrow\mathrm{72}\:{times} \\ $$$${x}\mathrm{3}{y}\:\Rightarrow\mathrm{8}×\mathrm{9}=\mathrm{72}\:{numbers}\Rightarrow\mathrm{72}\:{times} \\ $$$$\mathrm{3}{xy}\:\Rightarrow\mathrm{9}×\mathrm{9}=\mathrm{81}\:{numbers}\Rightarrow\mathrm{81}\:{times} \\ $$$${x}\mathrm{33}\:\Rightarrow\mathrm{8}\:{numbers}\Rightarrow\mathrm{16}\:{times} \\ $$$$\mathrm{33}{x}\:\Rightarrow\mathrm{9}\:{numbers}\Rightarrow\mathrm{18}\:{times} \\ $$$$\mathrm{3}{x}\mathrm{3}\:\Rightarrow\mathrm{9}\:{numbers}\Rightarrow\mathrm{18}\:{times} \\ $$$$\mathrm{333}\:\Rightarrow\mathrm{1}\:{numbers}\Rightarrow\mathrm{3}\:{times} \\ $$$$ \\ $$$$\Sigma{times}=\mathrm{1}+\mathrm{8}+\mathrm{9}+\mathrm{2}+\mathrm{72}+\mathrm{72}+\mathrm{81}+\mathrm{16}+\mathrm{18}+\mathrm{18}+\mathrm{3} \\ $$$$\Sigma{times}=\mathrm{300} \\ $$

Answered by mr W last updated on 27/Nov/18

3xy: 10×10=100 times x3y: 9×10=90 times 3y: 10 times xy3: 9×10=90 times x3: 9 times 3: 1 time Σtimes=100+100+100=300

$$\mathrm{3}{xy}:\:\mathrm{10}×\mathrm{10}=\mathrm{100}\:{times} \\ $$$$ \\ $$$${x}\mathrm{3}{y}:\:\mathrm{9}×\mathrm{10}=\mathrm{90}\:{times} \\ $$$$\mathrm{3}{y}:\:\mathrm{10}\:{times} \\ $$$$ \\ $$$${xy}\mathrm{3}:\:\mathrm{9}×\mathrm{10}=\mathrm{90}\:{times} \\ $$$${x}\mathrm{3}:\:\mathrm{9}\:{times} \\ $$$$\mathrm{3}:\:\mathrm{1}\:{time} \\ $$$$ \\ $$$$\Sigma{times}=\mathrm{100}+\mathrm{100}+\mathrm{100}=\mathrm{300} \\ $$

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