05.08.2019: Permutations and Combinations, 5/5: Pascal's Triangle and the Binomial Theorem (2/2)

Today's soundtrack is Deep Purple: Machine Head, an album that I bought because I loved playing "Highway Star" on Rock Band.

This morning, I'm continuing where I left off yesterday in learning about the binomial powers.

Now that we know how to expand a binomial power quickly using the concepts found in nCr, we can establish the Binomial Theorem, with which we can find the value of any term of an expanded binomial. The Binomial Theorem formula is

tₖ₊₁ = ₙCₖ aⁿ⁻ᵏ b

Note here that our term is k + 1; this means that if we want to find the value of the 5th term, we will need to substitute 4 for k to solve the equation.

At the right is an example of the binomial theorem in action.

Some imporant notes to make about expansions generated by the binomial theorem:

  • The sum of the exponents in each term is equivalent to the power of the binomial

  • The number of terms in the expanded form of a binomial power will be one more than the value of the binomial's exponent

  • A binomial raised to an even power will generate an equation with a middle term; an odd power will not have a middle term

  • We can find which term is the middle term (if there is one) by calculating ([n + 2] / 2), which means: two plus the value of the exponent, all divided by two.

If we are asked to find the constant term of a binomial, we must break it down algebraically and solve for k, finding the value of k that gives our variable the power of 0. See the example below:

That's it for today! Next time, I'll be starting on the final unit of this course, Function Notation and Operations.