# How to determine the difference between a problem that involves permutations and one that involves combinations?

Asked by ddude1116 (5575) May 16th, 2011

I know the difference between a permutation and a combination, permutations involve a specific order and combinations do not involve a specific order. But sometimes, it isn’t clear whether it’s a permutation or a combination. For instance: “Fifteen students ask to visit a college admissions counselor. Each visit includes one student. In how many ways can ten time slots be assigned?”, taken from my textbook. After some perusing, I think I may have figured out why it is a permutation, the book told me from the start, and that’s because it is one student, and the time element denotes an order. What are certain key things (i.e. time, quantity) to look for to distinguish this when math is out to get me?

Observing members: 0 Composing members: 0

That example is indeed tough. At first glance you’re not certain if the chronology of the meetings matters, or you’re just determining which ten students get to have meetings. But you’re absolutely right; since the question mentions meetings happening one at a time, chronology comes into play. Here are two very similar questions, but it will be clear which one is a combination and which is a permutation:

There were ten questions on last night’s homework assignment, and the teacher wants ten students to write the answers up on the board. If there are twenty students in the class, and no one student may do two questions, in how many ways can the students put the homework assignment on the board?

There were ten questions on last night’s homework assignment, and the teacher wants ten students to write the answers up on the board. If there are twenty students in the class, and no one student may do two questions, how many different groups of students can participate?

The difference is subtle, but the first is a permutation and the second a combination. The first specifies a difference between Mike doing problem #1 and Mike doing problem #9: if Mike does a different problem, that’s a different “way” in which the problems can be written on the board. The second only asks about the group of students participating versus the group of students not participating; it doesn’t matter which problem each student does.

Combinations generally focus on choosing a group, permutations focus on choosing a lineup.

Mariah (25876)

I hadn’t noticed that.. That makes a lot of sense, now that you mention it. Incredibly helpful answer! Thank you!

ddude1116 (5575)

or