Question: The number of times the digit 3 will be written when listing the integers from 1 to 1000 is
Options:
A) 269
B) 300
C) 271
D) 302
Show Answer
Answer:
Correct Answer: B
Solution:
- To find the number of times 3 occurs in listing the integer from 1 to 999. (since 3 does not occur in 1000). Any number between 1 to 999 is a 3 digit number $ xyz $ where the digit $ x,\ y,\ z $ are any digits from 0 to 9. Now, we first count the numbers in which 3 occurs once only. Since 3 can occur at one place in $ ^{3}C_1 $ ways, there are $ ^{3}C_1\ .\ (9\times 9)=3\ .\ 9^{2} $ such numbers. Again, 3 can occur in exactly two places in $ ^{3}C_1(9) $ such numbers. Lastly 3 can occur in all the three digits in one such number only 3337.
$ \therefore $ The number of times 3 occurs is equal to $ 1\times (3\times 9^{2})+2\times (3\times 9)+3\times 1=300 $ .
BETA
