In writing all of the integers from 1 to 300, how many times is the di

Lets go range wise 1- 100

Digit one can appear at any of the following places

Units , Tens and hundreds

Units
in a range of 1-100
1,11,21,31......91 i.e 10 times in a 100
so total occurence in 300 will be 3 *10 = 30

tens
10,11,12,13......19 =10times in a hundres
again total occurences in 300 will be 3 * 10= 30


hundreds
100 , 101, 102 ....109,....199 = 100 places

total = 30 +30 +100= 160

From 10 to 19, the digit 1 is used 11 times. For all other numbers from 1 to 99, it’s used a total of 9 times. So, it’s used a total of 20 times.

From 110 to 119, the digit 1 is used 21 times. For all other numbers from 100 to 199 it’s used 90 + 9 = 99 times. So, it’s used a total of 120 times.

From 210 to 219 the digit 1 is used 11 times. For all other numbers, from 200 to 300, it’s used 9 times, so it’s used a total of 20 times.

In total, the digit 1 is used 160 times.

Answer: D

Let's consider the values from 000 to 299
NOTE: Yes, I have started at 000 and ended at 299, even though though the question asks us to look at the numbers from 1 to 300.
HOWEVER, notice that 000 and 299 do not have any 1's so the outcome will be the same.

Okay, there are 300 integers from 000 to 299

UNITS position
Each of the 10 digits (0, 1, 2, 3..., 9) appears the same number of times in the UNITS position.
So, the digit 1 must occur 1/10 of the time in this position.
So, the digit 1 must appear (1/10)(300) times, which equals 30 times.

TENS position
Each of the 10 digits (0, 1, 2, 3..., 9) appears the same number of times in the TENS position.
So, the digit 1 must occur 1/10 of the time in this position.
So, the digit 1 must appear (1/10)(300) times, which equals 30 times.

HUNDREDS position
The 1 appears in the HUNDREDS position in every integer from 100 to 199
So, the digit 1 appears 100 times.

TOTAL = 30 + 30 + 100
= 160

Answer: D

Cheers,
Brent

There is no need to count! Here is a simple approach:

_ _ 1 : we have one fixed in the units, on the hundreds we have 1 to 3 to choose, that's 3c1, which is 3, and in the tens we have 0 to 9 to choose, 1 out of 10 digits, that's 10c1, which is 10. Hence, 3*10*1= 30

Move the 1 to the left:
_ 1 _ : we have one fixed in the tens, on the hundreds we have 1 to 3 to choose, that's 3c1, which is 3, and in the units we have 0 to 9 to choose, 1 out of 10 digits, that's 10c1, which is 10. Hence, 3*1*10= 30

Move the 1 to the left again:
1 _ _ : we have one fixed in the hundreds, in the units we have 0 to 9 to choose, 1 out of 10 digits, that's 10c1, which is 10. and in the tens we have 0 to 9 to choose, 1 out of 10 digits, that's 10c1, which is 10. Hence, 1*10*10= 100

Adding all up we get 160