Find Coefficient of x^9y^10 in (3x^3 - 4y^2)^8 | Combinatorics Problem Solution

[frankfjf]

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.

 

[Dick]

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.

 

[frankfjf]

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