2.2.3 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:

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

• 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 or 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