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 even and less than 200

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 even and less than 200

Answer 1

If the number must be even, our choices for the third number are 0, 2, 4, and 6.

If the number must be less than 200, our choices for the first number are 0 and 1.

So our possible combinations are of the form: 0_2, 0_4, 0_6, 1_0, 1_2, 1_4 and 1_6, where _ is the second digit. Note that 0_0 isn't allowed because we can't have repetition.

For each of the options above, there are 6 possibilities for the second digit. For 0_2, the second digit can be 1, 3, 4, 5, 6 or 7. For 0_4, the second digit can be 1, 2, 3, 5, 6, 7. And so on.

So there are 7 options for the first/third digit combination, each of which has 6 options for the second digit, for a total of 7 x 6 = 42 3-digit numbers that satisfy these conditions.