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

What do I need to know about combinations?

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