Expand a Trinomial Using Sigma Notation - 2 Examples

[dilan]

I find it difficult to expand a trinomial using the formula method (factorial method) where you can find the coefficient of any term without expanding the whole trinomial.
I can understand the binomial, but I can't do the trinomial using the general sigma notation method.
Can someone please show me how to do this by using about 2 examples?

Thanks a lot

 

[dilan]

anyone can help me? :(

 

[arildno]

Let your numbers be a,b,c. Define d=b+c. Then, we have:
$$(a+b+c)^{N}=(a+d)^{N}=\sum_{i=0}^{N}\binom{N}{i}a^{(N-i)}d^{i}=\sum_{i=0}^{N}\sum_{k=0}^{i}\binom{N}{i}\binom{i}{k}a^{(N-i)}b^{i-k}c^{k}$$

Denote the powers of a,b,c as $p_{a},p_{b},p_{c}$, respectively.

We therefore have that N,i and k are given by:
$$k=p_{c},i=p_{b}+p_{c},N=p_{a}+p_{b}+p_{c}$$
Thus, your coefficient, in terms of 3 powers are:
$$\binom{p_{a}+p_{b}+p_{c}}{p_{b}+p_{c}}\binom{p_{b}+p_{c}}{p_{c}}$$

seeing this pattern should tell you how to find the coefficients for higher nomials.