Permutation Combination & Probability

PERMUTATIONS

A permutation is "a re-arrangement of elements of a set".

So, what does this mean? It means a permutation is ONLY interested in re-arranging the elements of the set... Any duplications of the collected elements in different orders is fine.

A permutation therefore tends to be a large number.

Example:

Taking the 4 letters, ABCD, write down all the permutations of 3 of these leters:

ABC BAC CAB DAB

ACB BCA CBA DBA

ABD BAD CAD DAC

ADB BDA CDA DCA

ACD BCD CBD DBC

ADC BDC CDB DCB

There are 24 permutations in this case. In other words, just taking each letter and collecting them into sets of 3 from the 4 and writing them out, gives 24 variations. Done.

The order matters here, since ABC is different to ACB and different to BCA and different to CAB etc. Permutations see these as all different answers.

Mathematics of Permutations

To find the number of permutations of r elements from a set of n, the formula is:

So, the above example would be

COMBINATIONS

A combination is "one or more elements selected from a set without regard to the order"

The "without regard" means that the collection matters rather than order in combinations, so in the above example, the fact we ABC, ACB, BAC, BCA, CAB, CBA... for combinations, these are all 1 combination of letters A, B and C.

So, questions concerning picking a team of 5 people from a squad of 11... you would need combinations, since it is having "Bert, Ernie, Fred, Bill and Bob" that matters, not the fact that you have so many different permutations of these 5 people.

Example:

Taking the 4 letters, ABCD, write down all the combinations of 3 of these leters:

ABC ABD ACD BCD

  There are just 4 combinations in this case. You cannot pick any other 3 letters from ABCD, that is not part of the above 4 combinations. It is enlightening to see the letter missing in each: in order we have "no D", "no C", "no B" and finally "no A"... this sometimes helps you to "see" all the possible answers.

To calculate combinations, it is a 2-stage process:

1. you calculate all the equivalent permutations first.

2. you then correct this list by cutting out any duplicates.

As you can see from this, combinations are a subset of Permutations.

Mathematics of Combinations

To find the number of combinations of k elements from a set of n, the formula is:

So, the above example would be

EXAMPLE:

Q: From a squad of 16 members, find the total number of different arrangements of 9 players.

A: Now, the question, as usual has no mention of "permutation" or "combination", so we have to figure it out. Having read the above explanations now, hopefully you will appreciate that the question is one about combinations.

It is asking "find the number of combinations of 9 players from a squad of 16.

solution is:

use a calculator or work it out long handed:

with cancelling, we have

with further cancelling, we have which gives C(16,9) = 11440.