Apply counting principles, permutations, and combinations when presented.
Decide whether order matters before choosing between staged multiplication, permutations, or combinations.
核心知识
Use multiplication when choices happen in stages, permutations when order matters, and combinations when order does not matter. The first decision is not the formula; it is whether different orders count as different outcomes.
深入理解
Rule: ACT counting questions are usually short if you identify the structure immediately. If you are filling distinct positions, such as captain and vice-captain, order matters, so switching two people creates a new outcome. If you are just selecting a group, order does not matter, so the same members should not be counted more than once.
A reliable check is to ask, "Would swapping the order change the result in the real situation?" If yes, use a permutation or staged multiplication. If no, use a combination. This prevents the most common mistake: overcounting the same group many times.
分步讲解
- Decide whether the task is filling positions or choosing a group.
- If positions are different, multiply available choices by stage.
- If the same group can be listed in different orders, use combinations instead.
- Do a quick reasonableness check so the count is not too small or too large.
常见误解
- Using combinations when the problem has distinct roles.
- Using permutations when the order does not change the chosen group.
- Forgetting that the number of choices usually decreases after each selection when repetition is not allowed.
示例解析
A school has 8 finalists for student council. One student will be chosen as president and a different student will be chosen as vice-president. How many different outcomes are possible?
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