• question_answer The number of times the digit 3 will be written when listing the integers from 1 to 1000 is 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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