Finding Terms in a Binomial Expansion

Date: 03/27/2001 at 21:07:57
From: Jen
Subject: Finding terms for binomial expansion

I have a question on my homework that I really do not understand. I am 
asked to find the fifth term of:

     (-5m+4n^3)^54

I really have no idea how to solve this problem, and I was wondering 
if you could show me. I have no problem with doing problems like these 
up to the eighth power, but once it gets past that, I am totally 
clueless.

Please help.

Date: 03/27/2001 at 22:46:13
From: Doctor Roy
Subject: Re: Finding terms for binomial expansion

Hello Jen,

Thanks for writing to Dr. Math.

We can use the binomial theorem to solve this problem. The binomial 
theorem is related to Pascal's triangle, which you can also use to 
solve this problem. However, I do not think we wish to find the 54th 
row of the triangle, so let's use the theorem. The theorem states that 
the r-th term of the expansion of (a + b)^n is given by:

     C(n,(r-1)) * a^(n-(r-1)) * b^(r-1)

where C(n,r) = n! / (r!*(n-r)!)

So, for example, the 5th term in the expansion of (x - 2*y)^8 is given 
by:

       C(8,4) * a^(8-4) * b^4

     = [8! / (4!*4!)] * a^4 * b^4

     = 70 * x^4 * (-2*y)^4

     = 70 * x^4 * 16*y^4

     = 112 * x^4 * y^4

You can solve your problem in a similar fashion:

The fifth term of (-5m+4n^3)^54 is given by:

     C(54,4) * (-5*m)^(54-4) * (4*n^3)^4

The calculation is a bit tedious, but it can be done.

I hope this helps.

- Doctor Roy, The Math Forum
  http://mathforum.org/dr.math/