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Getting familliar with Counting methods

Problem Solving — algebra and arithmetic
Expert replies
by DBushkalov » Tue Feb 26, 2013 2:27 pm
hi guys,

after some tough hours with combinatorics and counting I am proud to finally state that i got to understand these topics a little better. Thus, the difference between permutations and combinations, as well as their respective use, are pretty much clearer to me.

However, I have one more questions:

when do we use only 7! or rather N! ?

It is clear to me that when you are counting the possible number of ways 6 people can sit next to each other you should calculate 6!. Or the examples wth the menus in a restaurant. But is there a general rule or a hint when to use only the N! ?

The way I understand it is:

you should proceed only with N! (namely, w/o dividing it by other factoriels) when you care for only 1 item of a set. one meal, one person etc. Thus, if you want to take 1 person from a group of ten, it is basically = 10! / ( 9! * 1! ), or simply 10. If you'd like to take the possible scenaris of getting 2 people out of a group, though, (w/o accounting for a particular order) the expression from above becomes: 10! / ( 8! * 2!), which is dfferent from 10 and that is why we write down the factoriels in the denominator.

It will be very helpful if someone could only tell me if i ve gotten this he right way, it is rather different.

thank you in advance and... off to probabilties. :))
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Source: — Quantitative Reasoning |

by ceilidh.erickson » Wed Feb 27, 2013 8:07 am
Yes, you seem to have a good understanding of how combinatorics work. Generally speaking, combinatorics/permutations problems fall into two categories:

Order doesn't matter: This will be the case when you're looking for GROUPINGS - teams, committees, mixtures, etc. In this case, ABC will be the same group as BAC, so you need to divide out the duplicates. As you said, if you're choosing groups of 2 people out of 10, it's 10!/(2!*8!). You divide by the factorial of what you're choosing (2) and what you're leaving out (8).

Order matters: This will be the case when you're looking for ARRANGEMENTS - seating arrangements, ways to arrange pictures on a wall, etc. When order does matter (when ABC is a different arrangement from BAC), you won't need to divide by any duplicates. So following your example, the number of ways to arrange 10 people in a row would simply be 10!. You could also think of this as 10!/(1!*1!*1!...). If you want to know how many ways to arrange 5 out of 10, it's 10!/(5!*1!*1!*1!*1!*1!) (5 people left out, then 1 for each of the other 5 positions), or 10*9*8*7*6.

So to answer your question, we simply use N! when order does matter, and we're arranging all of the possible items (putting 6 out of 6 people into a seating arrangement). If we're selecting certain people out of a larger group (6 out of 10 in seating arrangements), we need to divide by the factorial of the people we're not choosing (4!). If order doesn't matter, we also need to divide by the duplicates (with 6 out of 10 on a team, we divide by the 4! we're not including, and the 6! duplicates).

Does that clarify things?
Ceilidh Erickson
EdM in Mind, Brain, and Education
Harvard Graduate School of Education
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