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Permutations and Combinations

Posted: Sun Nov 03, 2013 4:05 am UTC
by free-bee
Over at a video game forum, someone asked how many different looks your character can have. I have not done these types of problems in two+ years. The numbers needed are currently unknown. So I will make up a problem from the top of my head which covers the question similarly.

Pretend you run a fan manufacturing business. Your fans can/will vary in terms of physical characteristics. Here are the possible options:
Fan Speed (x1000 RPM): 1, 2, 3
Number of Blades: 2, 3, 4, 5, 6, 7
Color: Pink, Black, Camouflage
Mist Sprayer: Yes/No

How many different fans can be produced?
I assume you count the options for each category and multiply them? If so, then the answer - in this specific question - is 3*6*3*2 = 108?

Re: Permutations and Combinations

Posted: Sun Nov 03, 2013 4:09 am UTC
by FancyHat
Yes.

Re: Permutations and Combinations

Posted: Sun Nov 03, 2013 1:57 pm UTC
by Flumble
To add to that: in general, if two choices are independent, you get the total number of possibilities by multiplying those of the individual choices.

Re: Permutations and Combinations

Posted: Mon Nov 04, 2013 1:35 am UTC
by dudiobugtron
Flumble wrote:To add to that: in general, if two choices are independent not mutually exclusive, you get the total number of possibilities by multiplying those of the individual choices.

Pedantically Robustened ;). You don't need independence to get the total number of possibilites.

Re: Permutations and Combinations

Posted: Mon Nov 04, 2013 10:42 am UTC
by DavCrav
I don't think that's clear. You do need independence of some sort, not just not mutually exclusive.

Here is an example, in the spirit of the OP: we'll take clothing and sex. There are (generically) two options for sex, but depending on the choice of sex, only certain clothing options might be available. You cannot simply multiply the (say) twnety clothing options by two sexes, even though sex and clothing are not mutually exclusive.

(You can decide whether this whole scenario was just a build up for that last line, a la http://xkcd.com/410.)

Re: Permutations and Combinations

Posted: Mon Nov 04, 2013 2:53 pm UTC
by jaap
DavCrav wrote:I don't think that's clear. You do need independence of some sort, not just not mutually exclusive.

Here is an example, in the spirit of the OP: we'll take clothing and sex. There are (generically) two options for sex, but depending on the choice of sex, only certain clothing options might be available. You cannot simply multiply the (say) twnety clothing options by two sexes, even though sex and clothing are not mutually exclusive.

(You can decide whether this whole scenario was just a build up for that last line, a la http://xkcd.com/410.)

So in your example certain clothes choices are excluded for certain sexes (and vice versa), i.e. those particular choices are mutually exclusive.

The problem with using 'independence' is that choices and combinations are often used in probability theory, and independence in that context is a far stronger requirement than needed if you only want to count all the combinations with non-zero probability (i.e. those that are not mutually exclusive).

Re: Permutations and Combinations

Posted: Mon Nov 04, 2013 10:53 pm UTC
by DavCrav
Hmm. So independent and mutually exclusive look like the same thing in this case. In which case, it isn't robustened, or whatever somebody earlier said. If they are different things, what is the difference in the context of combinations?

Re: Permutations and Combinations

Posted: Tue Nov 05, 2013 12:22 am UTC
by dudiobugtron
I think by attempting to 'robusten' it, I just made it ambiguous.

Here's an example that might help clear up what I meant.

Imagine you have two pairs of jeans (red and blue), and two tops (red and blue). You have to decide which ones to wear. You like colour-matching, so if you want to wear a blue top, that means you're more likely to also wear blue jeans. So the choices are not independent. However, you still might sometimes choose to wear the blue top with red jeans; if your blue jeans were in the wash, for example. So, they aren't mutually exlclusive.

There are four possible combinations in this setup RR, BB are the two most likely, but also RB and BR are possible.

However, if you never wore the red jeans with the blue top (because you hated that combination), then those two choices are mutually exclusive. So it's no longer one of the possible options.

As you can see, whether or not two choices are independent doesn't affect which combinations are possible, it just affects the chance of choosing those combinations. You only need to worry about whether they are mutually exclusive.