Hundereds Digit, Tens Digit, Ones Digit

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The numbers that are odd and do not contain the digit 5:

The hundreds digit can be any digit from 1 to 9 but it cannot be 5. So there are a total of 8 possibilities.

The tens digit can be any digit from 0 to 9 but it cannot be 5. So there are a total of 9 possibilities.

The ones digit must be odd so it can be 1, 3, 7, or 9 (it cannot be 5). So there are a total of 4 possibilities.

8 x 9 x 4 = 288

Total no of Odd integers between 100 and 1000 = 450.

No of odd integers containing 5:
First, start with 501-599. total no of odd integers containing 5 = 50.

Next take number between 100-200.
Total number of odd integers containing 5 = 14 ( 5 between 150-159 and 10 between 100-200, subtract 1 as we have double counted 155).
So between 100-1000, total number of odd integers containing 5 excluding 500-599 = 14 * 8 = 112.

Therefore, total number of odd integers without 5 digit = 450-50-112 = 288.
Ans (D)