JEE Main & Advanced Mathematics Permutations and Combinations Question Bank Critical Thinking Questions

  • 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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