# Combinatorics Problem

## Homework Statement

Find the coefficient of $$x^{9}$$ $$y^{10}$$ in (3$$x^{3}$$ - 4$$y^{2}$$)$$^{8}$$

## Homework Equations

The professor gave us a somewhat algebraic tactic or shortcut for solving these kinds of problems, mainly consisting of solving for each exponent. It can be somewhat tricky for me to explain with typing, but...

## The Attempt at a Solution

As per her suggested method, this is what I obtained:

C(8, k) (3$$x^{3}$$)$$^{8-k}$$ * ($$-4^{2}$$)$$^{k}$$

3(8 - k) = 9, 2k = 10. Thus k = 5 and 8 - k = 3.

This gives me the answer C(8,5) * $$3^{3}$$ * $$(-4)^{5}$$.

But her answer key in the provided review paper claims the answer is actually

C(8,3) * $$3^{3}$$ * $$(-4)^{5}$$

Have I done something wrong or is her answer incorrect? My text doesn't support her method and she is currently unavailable to assist me.

Last edited:

## Answers and Replies

Dick
Science Advisor
Homework Helper
C(8,3)=C(8,5). If you swap k and 8-k you get the key answer. They are both equal and both right.

Checked it out, you are correct. Thank you for your assistance.