This
is an interesting question from Counting. Simple framework, but one needs to be
very careful with the enumeration. One can get wrong answers in a number of
ways.
Question
If we listed all numbers from 100 to 10,000, how many times would the
digit 3 be printed?
A. 3980
B. 3700
C. 3840
D. 3780
Correct Answer
Choice A
Explanatory Answer
We need to consider all three digit and all 4-digit
numbers.
Three-digit numbers: A B C. 3 can be printed in the
100’s place or10’s place or units place.
Ø 100’s
place: 3 B C. B can take values 0 to 9, C can take values 0 to 9.
So, 3 gets printed in the 100’s place 100 times
Ø 10’s
place: A 3 C. A can take values 1 to 9, C can take values 0 to 9.
So, 3 gets printed in the 10’s place 90 times
Ø Unit’s
place: A B 3. A can take values 1 to 9, B can take values 0 to 9.
So, 3 gets printed in the unit’s place 90 times
So, 3 gets printed 280 times in 3-digit numbers
Four-digit numbers: A B C D. 3 can be printed in
the 1000’s place, 100’s place or10’s place or units place.
Ø 1000’s
place: 3 B C D. B can take values 0 to 9, C can take values 0 to 9,
D can take values 0 to 9. So, 3 gets printed in the 100’s place 1000 times.
Ø 100’s
place: A 3 C D. A can take values 1 to 9, C & D can take values
0 to 9. So, 3 gets printed in the 100’s place 900 times.
Ø 10’s
place: A B 3 D. A can take values 1 to 9, B & D can take values
0 to 9. So, 3 gets printed in the 10’s place 900 times.
Ø Unit’s
place: A B C 3. A can take values 1 to 9, B & C can take values
0 to 9. So, 3 gets printed in the unit’s place 900 times.
3 gets printed 3700 times in 4-digit numbers.
So, there are totally 3700 + 280 = 3980 numbers
Answer Choice (A)