Combinations (CIE A Level Maths: Probability & Statistics 1)

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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 begin mathsize 16px style P presuperscript n subscript r equals fraction numerator n factorial over denominator open parentheses n minus r close parentheses factorial end fraction end style
    • This would include r ! ways of repeating each combination
    • The formula for r combinations of n items is begin mathsize 16px style fraction numerator P presuperscript n subscript r over denominator r factorial end fraction equals fraction numerator n factorial over denominator open parentheses n minus r close parentheses factorial r factorial end fraction end style
  • The function begin mathsize 16px style fraction numerator n factorial over denominator open parentheses n minus r close parentheses factorial r factorial end fraction end style can be written as begin mathsize 16px style C presuperscript n subscript r end style or begin mathsize 16px style open parentheses table row n row r end table close parentheses end style  and is often read as ‘n choose r’
  • Make sure you can find and use the size 16px C presuperscript size 16px n subscript size 16px rbutton on your calculator
  • The formulae for permutations and combinations satisfy the following relationship:

Error converting from MathML to accessible text.

What do I need to know about combinations?

  • The formula begin mathsize 16px style C presuperscript n subscript r equals fraction numerator n factorial over denominator left parenthesis n minus r right parenthesis factorial r factorial end fraction end style is also known as a binomial coefficient
  • size 16px C presuperscript size 16px n subscript size 16px n size 16px equals size 16px C presuperscript size 16px n subscript size 16px 0 size 16px equals size 16px 1
    • 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 size 16px C presuperscript size 16px n subscript size 16px r size 16px equals size 16px C presuperscript size 16px n subscript size 16px n size 16px minus size 16px r end subscript
    • This can be seen by considering the formula for size 16px C presuperscript size 16px n subscript size 16px r
    • size 16px C presuperscript size 16px n subscript size 16px n size 16px minus size 16px r end subscript size 16px equals fraction numerator size 16px n size 16px factorial over denominator begin mathsize 16px style stretchy left parenthesis n minus r stretchy right parenthesis end style size 16px factorial size 16px left parenthesis size 16px n size 16px minus size 16px left parenthesis size 16px n size 16px minus size 16px r size 16px right parenthesis size 16px right parenthesis size 16px factorial end fraction size 16px equals fraction numerator size 16px n size 16px factorial over denominator begin mathsize 16px style r factorial left parenthesis n minus r right parenthesis factorial end style end fraction size 16px equals size 16px C presuperscript size 16px n subscript size 16px r

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

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?

2-2-3-combinations-we-solution

Examiner 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.

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Amber

Author: Amber

Expertise: Maths

Amber gained a first class degree in Mathematics & Meteorology from the University of Reading before training to become a teacher. She is passionate about teaching, having spent 8 years teaching GCSE and A Level Mathematics both in the UK and internationally. Amber loves creating bright and informative resources to help students reach their potential.