# [SOLVED] Trinomial expansion theorem

• Sep 2nd 2007, 05:54 AM
Shabir
[SOLVED] Trinomial expansion theorem
Binomial expansion theorem is well established, and technique to determine rth term for any binomial expansion function is also known. How to find precisely rth term in a trinomial expansion function: more specifically how to find 2nd,100th and 1600th term of (a-b-c)^125
• Sep 2nd 2007, 06:33 AM
ThePerfectHacker
Quote:

Originally Posted by Shabir
Binomial expansion theorem is well established, and technique to determine rth term for any binomial expansion function is also known. How to find precisely rth term in a trinomial expansion function: more specifically how to find 2nd,100th and 1600th term of (a-b-c)^125

You can think of trinomials as $\displaystyle (x+y+z)^{n} = [(x+y)+z]^{n} = \sum_{k=0}^{n}(x+y)^{n-k}z^k= \sum_{k=0}^{n} \left( \sum_{i=0}^{n-k}x^{n-k-i}y^i \right) z^k$.
• Sep 2nd 2007, 07:59 AM
Opalg
Quote:

Originally Posted by Shabir
Binomial expansion theorem is well established, and technique to determine rth term for any binomial expansion function is also known. How to find precisely rth term in a trinomial expansion function: more specifically how to find 2nd,100th and 1600th term of (a-b-c)^125

You can use the multinomial theorem to find any given term in such an expansion.

However, there is no natural total ordering of the terms in a multinomial expansion. You have to specify how you are going to order the terms before it makes sense to talk about the 2nd, 100th or 1600th term of (a-b-c)^125.