Question

The number of times the digit $5$ will be written when listing the integers from $1$ to $1000$, is

• A. 271
• B. 272
• C. 300
• D. None of these

Answer 1

Answer: (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 $xyz,0\le x,y,z\le 9$.
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 $=\left({}^3{C}_2\cdot 9\right)=27$
The number in which $5$ occurs in all three digits $=1$.
Hence, the number of times $5$ occurs $=1\times 243+2\times 27+3\times 1$
$=300$