See explanation.

Permutation of a set is a sequence in which the order is important (it means that for example ##135## and ##153## are two different permutations of a set ##{1,3,5}##)

Combination of a set is a subset in which the order is not important. This means that for example in a set ##{1,2,3,4,5,6}## you can find a 3-element subset ##{2,3,5}## and if you write it in decreasing order (as ##{5,3,2}##) it is still the same combination but written in a different way.

A problem including permutations would require ordering some elements. It can be a problem like:

Jack has a set of four books. Each is marked with a number 1,2,3 and 4. If he puts them randomly on a shelf, what is the probability that they are ordered either increasingly or decreasingly

Solution

To calculate the probability we have to calculate all possibilities first. There are 4 books and to find all possibilities of putting them on a shelf we have to calculate number of permutations of four element set:

##bar(bar(Omega))=P_4=4! =1*2*3*4=24##

The orders which are mentioned in the task are ##1234## (increasing order) and ##4321## (decreasing order), so

##bar(bar(A))=2##

Finally we can calculate the probability:

##P(A)=bar(bar(A))/bar(bar(Omega))=2/24=1/12##

A problem which includes combinations would deal with choosing a subset from a larger set. For example:

Before a science contest a team has to be chosen from a class. Assuming that there are 15 boys and 10 girls in how many ways can a team be choosen if it has to contain 2 girls and two boys ?

Solution:

In this task the order is not important. No matter if you choose Jack first and Ann next or the other way Ann first and next Jack, the team consists of the same people. So you have to use combinations to solve this task.

To calculate the number we can assume that we choose girls first, then boys, so the number would be calculated as:

##n=C””_10^2*C””_15^2=(10!)/(2!*8!)(15!)/(2!*13!)##

##n=15750##

Answer There are ##15750## ways to choose the team.