Choose With Restrictions

Choose With Restrictions

These exercises ask for a choice count when not every item or person is allowed. The main idea is to count only the valid selections, then simplify the result.

1) Identify the restriction

Read the condition carefully. Ask:

• Which objects are allowed?
• Are some choices excluded because of a rule?
• Does the restriction affect one item, several items, or the whole selection?

2) Count only valid choices

Work from the available options after applying the restriction. If the task involves selecting a group, count the number of ways to make that group using only acceptable choices. Keep the counting consistent with the wording of the problem.

3) Simplify the final answer

Write the result in its simplest exact form. If a fraction or expression appears, reduce it completely. If the answer is a count, make sure it is a whole number and that no extra factors remain.

4) Check your work

A good check is to verify that your count is not larger than the total number of unrestricted choices. Also confirm that every counted case satisfies the restriction and that no invalid case was included.

If you are unsure, rewrite the problem in your own words and list the acceptable options before counting.