The number of times the digit 5 will appear while writing the integers from 1 to 1000 is

Answer is (b). From 1 to 1000, the numbers in which 5 can occur could be of one digit, two digits or three digits. Case I – If the number is of one digit – 5 will appear only one time, i.e. in 5. Case II – If the number is of two digits – then (a) There is only one 5, this can happen in two ways _5 and 5_. In the first case (_5) the blank Place can be filled in 8 ways(as 0 and 5 cannot appear at that place), while in the second case (5_) the blank place can be filled in 9 ways (5 cannot appear there). Total 9 + 8 = 17 ways. (b) There are two 5s. In this case only ONE possibility. Case III – If the number is of three digits – then (a) Only one 5. Then, 5 can occupy three positions. 5 _ _ or _ 5 _ or _ _ 5. In the first case (5_ _), remaining two positions can be filled in 9 way each. So total 9 × 9 = 81 possibilities. In the second case (_ 5 _) first position can be filled in 8 ways and last position can be filled in 9 ways. So total 9 × 8 = 72 possibilities. Same will be true for the third (_ _ 5) case. So total 72 possibilities. (b) Only two 5. This can be done in three ways 55_ or 5_5 or _55. In first (55_) and second (5_5) case it can be filled in 9 ways each. While in the third case (_55) it can be filled in 8 ways. So total 9 + 9 + 8 = 26 possibilities. (c) All three digits are 5. This can be done in only ONE way. i.e, 555. So, total = 1 + 17 + 1 + 81 + 72 + 72 + 26 + 1 = 271.