A) 269
B) 300
C) 271
D) 302
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 \[1\times (3\times {{9}^{2}})+2\times (3\times 9)+3\times 1=300\].
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