# Coefficient and expansions.

• Feb 28th 2013, 01:40 AM
mshou
Coefficient and expansions.
find the Coefficient of term x2y3 in below expansion:

expansion : (2+x-y+3z)10

please write a good description for solution* or how to solve it?
* for a person who is a dummy in math and combinatorics...
• Feb 28th 2013, 02:09 AM
jakncoke
Re: Coefficient and expansions.
80,640. Use the multinomial theorem which says $(x_1+...+x_r)^n = \sum_{P(n)}{n \choose k_1,k_2,..,k_r } * (x_1)^{k_1}...(x_n)^{k_r}$(Basically sum over all non negative integer partitions of n.)
the ${n \choose k_1,k_2,..,k_r} = \frac{n!}{k_1!...k_r!}$
basically, it says the powers on the exponents of each term you have in your parenthesis, namely 2,x,y,z must add up to 10.
so how you can write 10 = 0+0+3+7 which is equivalent to $\frac{10!}{0!0!3!7!}2^0*x^0*y^3*z^7$
so since you asked for $x^2y^3$ this means since we don't see z so the power of z is 0. 2+3+0+x = 10 and so x = 5 (which is the power 2 is raised to)
so you have $\frac{10!}{0!5!3!2!}2^5*x^2*y^3*z^0$ = 80640 $x^2y^3$
• Feb 28th 2013, 02:25 AM
MINOANMAN
Re: Coefficient and expansions.
It is - 80640 x^2y^3

MINOAS
• Feb 28th 2013, 02:28 AM
mshou
Re: Coefficient and expansions.
well. thanks and a question:
isn't there any difference between for instance 3x and x, and -y and y ?
• Feb 28th 2013, 02:43 AM
mshou
Re: Coefficient and expansions.
well. thanks and a question:
isn't there any difference between for instance 3x and x, and -y and y ?