Number Properties Puzzle – Challenge 11

Challenge:

How many positive integers less than 1000 do not have any 5 as any digits?

A- 700

B- 728

C- 736

D- 770

E- 819

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Let’s consider 1 digit, 2 digits, and 3 digits numbers separately:
One digit number: From 9 1-digit numbers, eight numbers don’t have any digit as 5.
Two digits numbers: Put A for \(10^{th}\) place and B for unit place. Then, we have AB as a 2 digits number. We can put 8 digits for A (all digits except 0 and 5) and 9 digits for B (except 5). So, we’ll have 8 × 9 = 72 2-digits numbers that don’t have 5 as any digits.
For three digits numbers, ABC: For A, we can put 8 digits (not 0 or 5), for B, we have 9 digits (not 5) and for C, we have 9 digits (not 5). So, there are 648 three digits numbers that don’t have 5 as any digits. 8 × 9 × 9 = 648
In total: 8 + 72 + 648 = 728