## Question

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

### Solution

300

Since 3 does not occur in 1000, we have to count the number of times 3 0ccurs when we list the integers from 1 to 999. Any number between 1 and 999 is of the form 0 ≤ *x*, *y*, *z* ≤ 9. Let us first count the number in which 3 occurs exactly once. Since 3 can occurs at one place in ^{3}C_{1}ways, there are ^{3}C_{1 }(9 × 9) = 3 × 9^{2} such numbers. Next, 3 can occur in exactly two places in (^{3}C_{2})(9) = 3 × 9 such numbers. Lastly, 3 can occur in all three digits in one number only.

Hence, the number of times 3 occur is

.

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