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# How many number of times the digit '5' appear in listing the integers from 1 to 1000?

$\begin{array}{1 1} 271 \\ 272 \\ 280 \\ 300 \end{array}$

5 does not occur in 1000.
$\therefore$ we have to find the no. of times 5 occur in integers 1 to 999.
Integers from 1 to 999 are of the form $xyz$ where $0\leq x,y,z\leq 9$
case (i) Numbers in which 5 occur only once:
$=^3C_1\times 9\times 9=243$
(since 5 can be in unit's place or hundred's place or thousand's place.)
Case (ii) Numbers in which 5 occur twice :
$=^3C_2\times 9=27$
Case (iii) Numbers in which 5 occur thrice= 1
$\Rightarrow\:$ Number of times 5 occur $= (1\times 243)+(2\times 27)+(3\times 1)=300$