How many 5-number license plates can be made using the digits 1, 2, 3, 4, 5, 6, 7, if an odd digit must come first and a. repetitions are allowed b. repetitions are not allowed?

For Part A, there are 9,604 possibilities. For Part B, there are 1,440 possibilities.

To find the total number of possibilities, we will use the Fundamental Counting Principle. To do this, we find the number of possibilities for each position and then multiply them.

For Part B:
There are only 4 possibilities for the first position, since it must be odd (1, 3, 5, 7). The for the next 4 digits, all 7 numbers are a possibility.

4 x 7 x 7 x 7 x 7 = 9604

For Part B:
We again start with 4, however the numbers can be repeated. So we reduced the number of choice by 1 each time.
4 x 6 x 5 x 4 x 3 = 1440