how many different 3 digit numbers can be made from the digits 0 to 7 inclusive if repetition is not aloud and the number must be odd and greater than400

Question

how many different 3 digit numbers can be made from the digits 0 to 7 inclusive if repetition is not aloud and the number must be odd and greater than400

Answer 1

Since the number must be greater than 400, the first digit must be 4, 5, 6 or 7.

Since the number must be odd, the third digit must be 1, 3, 5 or 7.

Since repetition isn't allowed, our 3-digit options are of the form:

4_1, 4_3, 4_5, 4_7, 5_1, 5_3, 5_7, 6_1, 6_3, 6_5, 6_7, 7_1, 7_3, 7_5

For each of the above options, there are six options for the middle digit. For 4_1, the middle digit can be 0, 2, 3, 5, 6 or 7. For 4_3, the middle digit can be 0, 1, 2, 5, 6 or 7. And so on.

There are 14 possible first/last digit combinations, each of which has 6 middle digit options, for a total of 14 x 6 = 84 different 3-digit numbers that satisfy the required conditions.