This topic is about finding the numbers of distinct arrangements or selections from a given number of objects. Where the order of the objects is important permutations are used. If the order of the objects is not important then combinations are used.
Some calculators can evaluate the formulae for permutations and combinations.
A table of combinations, called binomial coefficients, for values from 0 to 15 objects, is available for Bursary examinations.
Factorials
A reminder that the factorial of a number n, is the product of all of the whole numbers up to and including n.
n! = n .(n − 1).(n − 2).(n − 3) ... 4.3.2.1 where n is a natural number |
Permutations
A permutation is an arrangement of elements from a set where the order of the elements is considered.
e.g. There are 6 possible permutations of two elements from the set {a, b, c}. These are (ab, ac, ba, bc, ca, cb)
The formula to work out the number of permuations of r elements from a set of n elements written is:
Example
How many permutations are there of three of the letters from the set {a, b, c, d, e}?
Combinations
A combination is an arrangement of elements from a set, where the order of the elements chosen is not considered .
e.g. There are only 3 possible combinations of two elements from the set {a, b, c}. These are (ab, bc, ac)
The formula to work out the number of combinations of r elements from a set of n elements written is:
The notation can also be written .
Example
How many combinations are there of three of the letters from the set {a, b, c, d, e}?
Permutations and Combinations with Restrictions
Sometimes problems involving permutations and combinations have restrictions or conditions.
Example
A mixed team of 5 players is to be chosen from a group of 6 boys and 7 girls.
(i) If the team can contain any numbers of boys and girls how many different teams could be chosen?
The number of combinations of 5 players chosen from a total of 13 players is required.
(ii) If the only restrictions on the team were that one particular boy and one particular girl had to be chosen, how many different teams could be picked?
In this problem only 3 players need to be found from the remaining 11.
(iii) If the team had to contain three girls and two boys, how many teams could be chosen?
The requirement now is to pick 3 girls from 7 girls and 2 boys from 6 boys.
The numbers from each selection are then multiplied together.