Binomial combination

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

Suppose we are expanding the expression (a+b)^2, into a^2 + 2ab + b^2.

THe coefficient of the ab term is 2. The reason for that is because, when we multiply out all the brackets, there are two times where we multiply a and b together:

• a from the first bracket and b from the second bracket
• b from the first bracket and a from the second bracket

That means that the binomial combinations for the ab term are 2.

We write that as ^2C_1.

That’s what ^nC_r means!

On a calculator

There’s a button on a calculator to calculate ^nC_r. 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 n value before it and the r value after it, e.g. 6 C 3 to calculate ^6C_3.

Casio CW calculator rant over.

Finding the coefficient of a term

Let’s say we want to find the coefficient of the a^2b^3 term of the expansion of (a+b)^5. That would be ^5C_2 because the a^2b^3 term is the term where a has power 2.

Terms with coefficients

If we want to find the coefficient of the x^3 term of the expansion of (2+3x)^5:

• Find ^5C_3:
  • ^5C_3=10
• The extra coefficients are:
  • (3x)^3 = 27x^3
  • 2^2 = 4
• The extra coefficients are 4 \times 27x^3 = 108x^3
• If we times everything together, the coefficient of the x^3 term is 10 \times 108 = 1080.

Find the coefficient of the x^4 term of the expansion of (1+2x)^6.

• ^6C_4=15
• The extra coefficients are:
  • (2x)^4 = 16x^4
  • 1^2 = 1
• The extra coefficient is 1 \times 16x^4 = 16x^4
• If we times everything together, the coefficient of the x^4 term is 15 \times 16 = 240.
• Answer: 240.

Find the first 3 terms of the expansion of (3+x)^5.

• For the x^0 term:
  • ^5C_0=1
  • The extra coefficients are:
    • (3)^5 = 243
    • x^0 = 1
  • The extra coefficient is 243 \times 1 = 243
  • The coefficient of the x^0 term is 1 \times 243 = 243.
• For the x^1 term:
  • ^5C_1=5
  • The extra coefficients are:
    • (3)^4 = 81
    • x^1 = x
  • The extra coefficient is 81 \times x = 81x
  • The coefficient of the x^1 term is 5 \times 81 = 405.
• For the x^2 term:
  • ^5C_2=10
  • The extra coefficients are:
    • (3)^3 = 27
    • x^2 = x^2
  • The extra coefficient is 27 \times x^2 = 27x^2
  • The coefficient of the x^2 term is 10 \times 27 = 270.
• Answer: 243 + 405x + 270x^2 + ....