I'm currently studying introductory statistics, and I'm having a difficult time intuitively understanding the approach here.

In my textbook, it says: "Selecting a sample of three persons from a group of six people to form a committee of three people results in the choice of 20 committees." This is because "6 choose 3" equals 20.

But next it says, "If we were instead to select a treasurer, a captain, and a typist out of the group of six, we would have a total of 6 x 5 x 4 = 120 outcomes."

I am having a hard time understanding why these are two different problems. Aren't they both asking to choose three people from a list of six? Does it have to do with order, or replacement?


It is a situation equivalent to ordering.

If instead of calling the positions Treasurer and Captain and Secretary, you called them One and Two and Three, and assigned them by the order they were pulled, the relationship is clear.

To further help make sense of it, reduce the numbers in the problem and solve that instead.

Picking Alex as treasurer and Bobby as captain is not the same thing as picking Bobby as treasurer and Alex as captain.

However, picking Alex first and Bobby second makes a committee of Alex and Bobby, which is the same thing as picking Bobby first and Alex second.

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  • $\begingroup$ Thank you for this explanation! That makes sense now. "6 choose 3" doesn't care about the order, hence there are less outcomes because ABC is the same as CBA. But when choosing the individual roles, ABC and CBA become two separate outcomes. That makes a lot of intuitive sense--thank you again! $\endgroup$ – SwordSaintAshina Oct 29 at 8:53
  • $\begingroup$ @SwordSaintAshina Incidentally you might notice that 120 = 20 * 6; once you have selected a team of 3, there are 6 different ways to assign these 3 people to the 3 posts Treasurer, Captain and Typist. Generally, the number of ways to assign k people to k posts out of n people is (n choose k) * n!. $\endgroup$ – Stef Oct 29 at 14:09

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