Question

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

Answer 1

Correct option (b)

Explanation:

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 xyz where 0 ≤ x, y, z ≤ 9. Let us first count the numbers in which 3 occurs exactly once. Since 3 can occur at one place in ³C₁ ways, there are ³C₁ (9 × 9) = 3 × 9² such numbers. Next, 3 can occur in exactly two places in (³C₂) (9) = 3 × 9 such numbers. Lastly, 3 can occur in all three digits in one number only. Hence, the number of times 3 occurs is 1 × (3 × 9²) + 2 × (3 × 9) + 3 × 1 = 300.