The Binomial Theorem is easier and more naturally proven in a combinatorics context but can be proven by induction.

**Problem: ***Prove the Binomial Theorem by Induction.*

**Solution: **Let be the proposition that for ,

(*Binomial Theorem*)

:

( is true)

Now assume is true; that is:

Now

Now all terms are of the form as runs from . Let . Now the term has constant from and :

It is a straightforward exercise to show:

Hence

( is true)

is true. . Hence is true for all ; i.e. the Binomial Theorem is true

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November 17, 2011 at 8:52 pm

Sinead HanafinHi This is Sinead Hanafin (Gurteens Annascaul) and I just have a quick question, I was wondering why you did binomial expansions as i asked my teacher and she didn’t answer

Thanks

November 17, 2011 at 9:13 pm

J.P. McCarthySinead,

There are four main uses of the Binomial Theorem for LC Maths that I can think of.

1. Expanding powers of Sums

Rather than having to multiply expressions of the type by hand we can use the Binomial Theorem. For example,

,

,

.

2. Proving that the power rule for differentiation (positive powers)

The normal approach is to use induction on the proposition

but you can also do it from first principles using the Binomial Theorem. See http://irishjip.wordpress.com/2011/03/29/leaving-cert-maths-differentiation-from-first-principles/

3. Identities involving the binomial coefficients

For example,

,

,

4. Probability: The Binomial Distribution

A simple example that you flip ten coins. What is the probability of getting six heads. We can show that this is given by

.

You’ll see more of this kind of stuff in Project Maths.

J.P.