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

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Since 3 does not occur in 1000, we have to count the number of times 3 occurs when we list the integers from 1 to 999. Any number between 1 and 999 is of the form $x yz$ where $0 \leq x, y, z \leq 9$ .

Let us first count the number in which 3 occurs exactly once. Since 3 can occur at one place in $^{3} C_{1}$ ways, there are $^{3} C_{1} (9 \times 9) = 3 \times 9^{2}$ such numbers. Next, 3 can occur in exactly two places in $^{3} C_{2} (9) = 3 \times 9$ such numbers. Lastly, 3 can occur in all three digits in one number only.

Hence, the number of times 3 occurs is:

$1 \times (3 \times 9^{2}) + 2 \times (3 \times 9) + 3 \times 1 = 300$

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