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

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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 $= (1\times 243)+(2\times 27)+(3\times 1)=300$

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