Binomial Coefficients & Pascal's Triangle (DP IB Analysis & Approaches (AA)): Revision Note

The Binomial Coefficient ⁿC_r

What is the formula for ?

• The formula for is is

• where the factorial symbol means, for example:

  • 

• e.g.

What properties of  do I need to know?

• Useful properties of to know are

  • 

  • 

    • note that

  • 

  • The numbers are symmetric

    • i.e.

How does  relate to the binomial theorem?

• The binomial theorem gives you the expansion of for different positive integer powers of :

• where is the binomial coefficient

How does  relate to counting principles?

• The value of represents the number of ways to choose objects out of different objects

  • This is an example of a counting principle

  • e.g. how many ways can you choose 2 people out of 5 people?

    • There are ways to do this

How is the binomial theorem related to counting principles?

• You can relate in the binomial theorem to the number of ways to choose objects out of different objects

• means multiply by itself times

  • i.e.

  • Without using the binomial theorem, expand the right-hand side

    • i.e. multiplying all combinations of taking a letter from each bracket

• e.g. one possibility is where the last 2 are 's

  • The first are 's

  • This simplifies to

  • But you can also choose 2 's out of another 2 brackets (not just the last 2)

    • How many ways can you chose 2 's out of the brackets?

  • This is number of ways to choose objects out of different objects

    • i.e. there are ways to do this

  • so the full term is

Pascal's Triangle

What is Pascal’s triangle?

• Pascal’s triangle is formed as follows:

  • The first row and second row form a triangle of 1s

  • Every row below starts and ends with a 1

  • The middle terms are found by

    • adding the two terms above it

How does Pascal’s triangle relate to binomial coefficients ?

• The binomial coefficients are rows in Pascal’s triangle

• e.g. calculate

  • If you use the formula

    • e.g. (as )

    • e.g.

    • etc

  • you get

  • This is the same as the row in Pascal's triangle!

How does Pascal’s triangle relate to the binomial theorem?

• The binomial theorem gives you the expansion of for different positive integer powers of :

• Instead of using to calculate

  • 

  • just read the values off the relevant row in Pascal's triangle!