• # question_answer The number of times the digit 5 will be written when listing the integers from 1 to 1000 is A) 271 B) 272 C) 300 D) None of these

Since 5 does not occur in 1000, we have to count the number of times 5 occurs when we list the integers from 1 to 999. Any number between 1 and 999 is of the form$xyz,\ 0\le x,\ y,\ z\le 9$. The numbers in which 5 occurs exactly once$={{(}^{3}}{{C}_{1}})\ .\ 9\times 9=243$. The numbers in which 5 occurs exactly twice =${{(}^{3}}{{C}_{2}}.9)=27$ The numbers in which 5 occurs in all three digits $=1$ Hence, the number of times 5 occurs is $1\times 243+2\times 27+3\times 1=300$.