N term in formula for combinations

In summary, the n! factor comes about because there are that many ways to choose n things from N. The number of "permutations" of N things, taken i at a time, and sometimes called NPi is N!/(N-i)!. The n! term fixes it up.
  • #1
mmwave
647
2
I once read a very good explanation for the n! factor in the combinations formula but I can't find it. Can someone state the reason for it clearly please.

( N) = N!/n!/(N-n)!
( n)

The N!/(N-n)! comes about because there are that many ways to choose n things from N. N for the first, N-1 for the second, ... N-n+1 for the nth. So far so good.

There is a clear explanation for why this has 'double counted' and the n! term fixes it up.
 
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  • #2
The number of "permutations" of N things, taken i at a time, and sometimes called NPi is N!/(N-i)!. That's true because you have N choices for the first thing, then N-1 choices for the second, etc. until you have taken i of them. That's N(N-1)(N-2)...(N-i+1)) That's a total of i factors and the last of them is
(N-i)+1. One way to write that is to start with N! and "cancel" those below N-i+1 by dividing by (N-1)!.

Look at an example. If we had 4 things: {a, b, c, d} and asked for "all permutations of those 4 things, take 2 at a time" we would be seeking {a,b}, {a,c}, {a,d} (I've chosen a first and then each of the other 3 second), {b,a}, {b,c}, {b,d} (choose b first), {c,a}, {c,b}, {c,d} (get the idea?), {d,a}, {d,b}, {d,c}.
A total of 4*3= 4!/2!= 4!/(4-2)!= 4P2 permutations.

That's the part you have.

Now, to get "combinations" where order is not important:

The 12 permutations we have include both {a,b} and {b,a}, {a,c} and {c,a}, etc. In fact every pair is "duplicated" in reverse order. That's because there are 2!= 2 ways of permuting 2 things.
If we don't want to count permutations as different combinations, we need to divide by 2: there are 6= (4P2)/2
= (4!/2!)/2!= 4!/((4-2)!2!)= 4C2.

If we want all combinations of N things taken i at a time, we can calculate, as above, that there are N!/(N-i)! permutations. Since there are i! ways to permute the same i things, to not count permutations of the same things as different combinations, we need to multiply by i!: N!/((N-i)!i!)= NCi.
 
  • #3
Thanks! That was it, we have ab and ba so divide by 2. If n things divide by n!

I knew it was not as complicated as my statistical physics teacher made it sound.
 

1. What is the N term in the formula for combinations?

The N term in the formula for combinations represents the total number of objects in a set.

2. How is the N term used in the formula for combinations?

The N term is used as the upper limit of the summation in the formula for combinations, indicating the number of objects to be chosen from the set.

3. Can the N term be a decimal or a negative number?

No, the N term must be a positive integer in the formula for combinations as it represents the number of objects in a set.

4. Is the N term the same as the number of combinations in a set?

No, the N term is not the same as the number of combinations in a set. It is used to calculate the number of combinations, but does not represent the actual number of combinations.

5. What is the difference between the N term and the R term in the formula for combinations?

The N term represents the total number of objects in a set, while the R term represents the number of objects to be chosen from the set. Together, they are used to calculate the number of combinations in a given set.

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