Question

The number of times the digit 5 will be written, when all the numbers from 1 to 1000 are written is

• A. 300
• B. 243
• C. 297
• D. None of these

Answer 1

Answer: (A)

300

Answer 2

Observe that 1000 does not contain the digit 5. While listing the numbers from 1 to 999, we have to write a sequence of three digits such as xyz where x, y, z ∈ {0, 1, 2, 3, 4, 5, 6, 7, 8,9}.

When 5 occurs only at one place (which can happen in $3C_{1} = 3$ ways) then the other two places can be filled in 9 x 9 ways. So, in this case 5 will be written 3 x 9 x 9 times.

($\because$5 may occur in any one of the 3 places).

When 5 occurs at two places (which can happen in $3C_{2} = 3$ ways) then the third place can be filled in 9 ways. So, there will be 3 x 9 = 27 such numbers and while writing any such number 5 will be written twice. So, in this case 5 will be written 2 x 27 times.

When 5 occurs at all the three places (which can happen in only one way), we have to write 5 three times while listing the only number so formed.

Hence, the required number of times 5 will be written

= 3 x 9 x 9 + 2 x 27 + 3

= 243 + 54 + 3 = 300