Coefficient of term in expansion

[Gwilim]

Homework Statement

Find the coefficient of X^3.Y^2 in (1+2X+5Y)^6

The Attempt at a Solution

This question would be much easier for me if not for the constant term inside the brackets. I seem to recall that the coefficients of expansion for functions such as (ax+by)^n can be determined by looking at the relative line of pascals triangle. One method of approaching the problem, in a more general sense since I need to be able to answer questions like this in exams, which I tried was expanding (1+x+y)^2 and comparing it with (x+y)^2, which produced an interesting result. it was quite clear though that doing the same for (1+x+y)^3 and above and trying to use induction to come up with a general rule for that kind of relationship would mean giving this type of problem far more time and attention than I can afford to, and I might be running up a blind alley anyway.

Another possibility that occurred to me was modelling (1+aX+bY) as a vector, but I'm kind of shaky on this.

Anyway I'm sure this is a very easy question for anyone familiar with the method required to solve it, so if anyone could point me in the direction of some instructions on how to do these problems, or give a brief explanation themselves I'd be very grateful.

 

[Dick]

Let A=(1+2x) and B=5y. So your expression is (A+B)^6. You can read off the coefficients of the expansion from Pascal's triangle, as you said. What's the coefficient of A^4*B^2? That's the one you want because you need a y^2 and no other term will give you one. Now you just have to figure out the coefficient of x^3 in (1+2x)^4...

 

[Gwilim]

Fantastic! Thankyou so much.