Question

In writing all of the integers from $1$ to $199$, how many times is $\text{'}1\text{'}$ used?

Answer 1

Step 1: Counting the number of times one occurs from $1$ to $99$

$1,21,31,41,51,61,71,81,91–9$ times in unit digit

$10,12,13,14,15,16,17,18,19–9$ times in ten’s Digit

$11-2$ times ,i.e., at unit and ten's digit

$9+9+2=20$ times

Step 2: Counting the number of times one occurs from $100$ to $199$

$111$will be counted for unit digit, ten's digit and hundred's digit so it will be come $3$ times.

$101,121,131,141,151,161,171,181,191–9$ times at unit digit

$110,112,113,114,115,116,117,118,119–9$ times at ten’s Digit

$100,102,103,104\dots \dots \dots \dots \dots 198,199-81$ times at hundred's digit.

$\Rightarrow 81\times 1+9\times 2+9\times 2+3=120$ times

Step 3: Number of times one occurs from $100$ to $199$

$120+20=140$

Therefore, in all integers from $1$ to $199$, the total number of times $\text{'}1\text{'}$ is used is $140$ times.