Combinations (DP IB Analysis & Approaches (AA)): Revision Note

Combinations

What is a combination?

• A combination is the number of ways to select objects out of different objects, where the order does not matter

  • e.g. selecting Ravi, Euan and Jess out of 10 possible students to work together in a small group is a combination

    • The order of ' Ravi / Euan / Jess' does not matter

What is the difference between a combination and a permutation?

• In a permutation the order matters

  • but in a combination, the order does not matter

• e.g. selecting 4 paints out of 9 different paints to mix together and form a new colour is an example of a combination

  • the order of the 4 paints does not matter

    • as they all get mixed together

• However selecting 4 paints out of 9 different paints to colour four different regions on a flag is an example of a permutation

  • the order matters

    • e.g. a flag starting with red looks different to a flag starting with blue

What is  ?

• stands for the number of ways to select objects out of different objects when order does not matter

  • and where

• The formula is

  • Note that it is not possible to select repeated objects when using

• e.g. how many teams of 4 people can be made out of 12 people?

  • Each of the 12 people are different

  • You are selecting out of

  • Order does not matter

    • A team with X, Y and Z in it is the same as a team with Z, Y and X in it

  • so

  • This can be simplified by cancelling

    • 

What properties of  do I need to know?

• Useful properties of to know are

  • 

  • 

    • note that

  • 

  • The numbers are symmetric

    • i.e.

How is the formula related to the  formula?

• and

  • so

• This means is times bigger than

• This is true because is the number of ways to select objects out of different objects where order matters

  • i.e. after you select objects out of

    • i.e.

  • you then find all the possible rearrangements of those objects

    • i.e. multiply by rearrangements

When do I multiply   values together?

• If asked to find combinations out of subgroups of the different objects, multiply together the values

• e.g. in a class of 20 students, 5 students are left-handed and the rest are right-handed. How many ways can a team of 6 students be formed in which 2 are left-handed?

  • Out of the 5 left-handed students, select 2

    • 

  • Out of the 15 right-handed students, select 4

    • 

  • Multiply these values together

    • ways