# Thread: [SOLVED] Trinomial expansion theorem

1. ## [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

2. 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 $\displaystyle (x+y+z)^{n} = [(x+y)+z]^{n} = \sum_{k=0}^{n}(x+y)^{n-k}z^k= \sum_{k=0}^{n} \left( \sum_{i=0}^{n-k}x^{n-k-i}y^i \right) z^k$.

3. 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.