Binomial combination

To understand what a binomial combination is, let’s do an example.

Suppose we are expanding the expression , into .

THe coefficient of the term is . The reason for that is because, when we multiply out all the brackets, there are two times where we multiply and together:

• from the first bracket and from the second bracket
• from the first bracket and from the second bracket

That means that the binomial combinations for the term are .

We write that as .

That’s what means!

On a calculator

There’s a button on a calculator to calculate . Often it’s above the divide button (press shift), but on some of the new ‘upgraded™®’ casio calculators you have to:

• Press the Catalog button
• Scroll down and click Probability
• Scroll down to Combination C and click it
• Then put the value before it and the value after it, e.g. 6 C 3 to calculate .

Casio CW calculator rant over.

Finding the coefficient of a term

Let’s say we want to find the coefficient of the term of the expansion of . That would be because the term is the term where has power .

Terms with coefficients

If we want to find the coefficient of the term of the expansion of :

• Find :
• The extra coefficients are:
• The extra coefficients are
• If we times everything together, the coefficient of the term is .

Find the coefficient of the term of the expansion of .

• The extra coefficients are:
• The extra coefficient is
• If we times everything together, the coefficient of the term is .
• Answer: .

Find the first 3 terms of the expansion of .

• For the term:
  • The extra coefficients are:
  • The extra coefficient is
  • The coefficient of the term is .
• For the term:
  • The extra coefficients are:
  • The extra coefficient is
  • The coefficient of the term is .
• For the term:
  • The extra coefficients are:
  • The extra coefficient is
  • The coefficient of the term is .
• Answer: .