First slide

Combinations

Question

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

Moderate

A

269
B

300
C

271
D

302

Solution

Any number between 1 to 999 is a 3,digit number xyz
where the digits xyz are any digits from 0 to 9.
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 $^{3} C_{1} \cdot (9 \times 9) = 3 \cdot 9^{2}$ such numbers
$3_{C_{1}} 9^{2} + 2 \times 3_{C_{2}} \times 9 + 3 \times 3_{c_{3}} = 243 + 54 + 3 = 300$

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