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

The Binomial Coefficient nCr

What is ?

• If we want to find the number of ways to choose r items out of n different objects we can use the formula for 

  • The formula for r combinations of n items is 

  • This formula is given in the formula booklet along with the formula for the binomial theorem

  • The function can be written  or and is often read as ‘n choose r’

    • Make sure you can find and use the button on your GDC

How does  relate to the binomial theorem?

• The formula  is also known as a binomial coefficient

• For a binomial expansion the coefficients of each term will be ,  and so on up to 

  • The coefficient of the  term will be 

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• The binomial coefficients are symmetrical, so 

  • This can be seen by considering the formula for 

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Pascal's Triangle

What is Pascal’s Triangle?

• Pascal’s triangle is a way of arranging the binomial coefficients and neatly shows how they are formed

  • Each term is formed by adding the two terms above it

  • The first row has just the number 1

  • Each row begins and ends with a number 1

  • From the third row the terms in between the 1s are the sum of the two terms above it

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

• Pascal’s triangle is an alternative way of finding the binomial coefficients, 

  • It can be useful for finding for smaller values of without a calculator

  • However for larger values of it is slow and prone to arithmetic errors

• Taking the first row as zero, , each row corresponds to the row and the term within that row corresponds to the term