Combinations
What is the difference between permutations and combinations?
 A combination is the number of possible arrangements of a set of objects when the order of the arrangements does not matter
 On the other hand a permutation is when the order of arrangement does matter
 A combination will be finding the number of ways to choose r out of n items
 The order in which the r items are chosen is not important
 For example if we are choosing two letters from the word CAB, AB and BA would be considered the same combination but different permutations
How do we find r combinations of n items?
 If we want to find the number of ways to choose 2 out of 3 different objects, but we don’t mind the order in which they are chosen, then we could find the number of permutations of 2 items from 3 and then divide by the number of ways of arranging each combination
 For example if we want to choose 2 letters from A, B and C
 There are 6 permutations of 2 letters:
 For example if we want to choose 2 letters from A, B and C
AB, BA, AC, CA, BC, CB


 For each combination of 2 letters there are 2 (2 × 1) ways of arranging them
(for example, AB and BA)  So divide the total number of permutations (6) by the number of ways of arranging each combination (2) to get 3 combinations
 For each combination of 2 letters there are 2 (2 × 1) ways of arranging them

 If we want to find the number of ways to choose 3 out of 5 different objects, but we don’t mind the order in which they are chosen, then we could find the number of permutations of 3 items from 5 and then divide by the number of ways of arranging each combination
 For example if we want to choose 3 letters from A, B, C, D and E
There are 60 permutations of 3 letters:
ABC, ACB, BAC, BCA, CAB, CBA, ABD, ADB, etc


 For each combination of 3 letters there are 6 (3 × 2 ×1) ways of arranging them (for example, ABC, ACB, BAC, BCA, CAB and CBA)
 So divide the total number of permutations (60) by the number of ways of arranging each combination (6 which is 3!) to get 10 combinations

 If we want to find the number of ways to choose r items out of n different objects, but we don’t mind the order in which they are chosen, then we could find the number of permutations of r items from n and then divide by the number of ways of arranging each combination
 Recall that the formula for r permutations of n items is
 This would include r ! ways of repeating each combination
 The formula for r combinations of n items is
 The function can be written as or and is often read as ‘n choose r’
 Make sure you can find and use the button on your calculator
 The formulae for permutations and combinations satisfy the following relationship:
What do I need to know about combinations?
 The formula is also known as a binomial coefficient

 It is easy to see that there is only one way of arranging n objects out of n and also there can only be one way of arranging 0 objects out of n
 By considering the formula for this, it reinforces the fact that 0! Must equal 1
 The binomial coefficients are symmetrical, so
 This can be seen by considering the formula for
How do I know when to multiply of add?
 Many questions will ask you to find combinations of a group of different items from a bigger group of a specified number of those different items
 For example, find the number of ways five questions could be chosen from a bank of twenty different pure and ten different statistics questions
 The hint in this example is the word 'chosen', this tells you that the order in which the questions are chosen doesn't matter
 Sometimes questions will have restrictions,
 For example there should be three pure and two statistics chosen from the bank of questions,
 Or there must be at least two pure questions within the group
 If unsure about whether to add or multiply your options, ask yourself if A and B are both needed, or if A or B is needed
 Always multiply if the answer is and, and add if the answer is or
 For example if we needed exactly three pure and two statistics questions we would find the amount of each and multiply them
 If we could have either five statistics or five pure questions we would find them separately and add the answers
 Probabilities can be found with combinations questions by finding the number of options a selection can be made in a particular way and dividing that by the total number of options
How do we handle combinations if some of the objects are identical?
 Sometimes you will be asked to find the number of ways r items can be chosen from n items when some of the items are identical
 You must consider the identical items separately
 For example, if you wanted to choose 4 letters from the word CHOOSE you would have to consider all the options with zero Os, one O and two Os separately
Worked Example
Oscar has to choose four books from a reading list to take home over the summer. There are four fantasy books, five historical fiction books and two classics available for him to choose from. In how many ways can Oscar choose four books if he decides to have:
(i)
two fantasy books and two historical fictions?
(ii)
at least one of each type of book?
(iii)
at least two fantasy books?
Exam Tip
 It is really important that you can tell whether a question is about permutations or combinations. Look out for key words such as arrange (for permutations) or choose or select (for combinations). Don’t be confused if a question asks for the number of ways, this could be for either a permutations or a combinations question. Look out for other clues.