How many odd three-digit integers greater than 800 are there such that their three digits are different?
A) 40
B) 56
C) 72
D) 81
E)104
A) 40
B) 56
C) 72
D) 81
E)104
IMO C.
Between 801 & 900, there are 100 digits of which 50 are even & 50 are odd. We need to subtract the numbers that have the units & tens digits same.
For the number 8 _ _ to be odd, the units digit should be odd (i.e. 1,3,5,7,9). The tens digit can be any number between 0-9, but should not be the same at the units & hundreds digit.
The odd units digit can be selected in 5 ways. The tens digit can then be selected in 9 ways. So, totally there are 5*9=45 odd numbers that don't have the same units & tens digit. Now, there are 881, 883, 885, 887 & 889 that were included in the count. So, totally there are 50-10 = 40 odd numbers between 800-900 that have all 3 digits different.
For the number 9_ _ to be odd, similar logic as above should be applied. There are 18 numbers that have either the hundreds-unit or hundred-tens or tens-units that have the same odd number. So, totally there are 50-18 = 32 odd numbers between 901-999 that have all 3 digits different.
Totally, there are 40+32 = 72 odd integers that have all 3 digits different.
