Explanation

(c)
Since, 5 does not occur in 1000, we have to count the number of times 5 occurs when we list the integers from 1 to 999. Any number between 1 and 999 is of the form $xyz,\ 0 \leq x,\ y,\ z \leq 9$
The number in which 5 occurs exactly once
$= \left( {\ ^{3}C}_{1} \right)9 \times 9 = 243$
The number in which 5 occurs exactly twice=$\left( {\ ^{3}C}_{2}.9 \right) = 27$
The number in which 5 occurs in all three digits =1
Hence, the number of times 5 occurs
$= 1 \times 243 + 2 \times 27 + 3 \times 1 = 300$