Counting with restrictions

Multiply the numbers of choices when each stage follows the previous stages. If digits cannot repeat, each used digit is removed from the remaining choices. Deal first with a position having a special restriction, such as an odd final digit or a nonzero first digit. The order of counting positions need not match their written order.

Separate cases when the number of remaining choices changes. For example, allowing a zero creates a leading-digit restriction which may depend on the final digit already chosen. Add disjoint cases; multiplying overlapping possibilities counts arrangements more than once. Listing a smaller version of the problem is a useful check.

Worked example

How many three-digit odd numbers can be formed from 1, 2, 4 and 7 without repeating a digit?

1. The final digit is 1 or 7: two choices.
2. There are three remaining choices for the first digit, then two for the middle digit.
3. Multiply 2 × 3 × 2.

Answer: 12