2.5 Binomial expansions (in class)

What is the coefficient of [math]x^3y^3[/math] in the expansion of [math](x-2y)^6[/math]?

(Calculator) Consider a function [math]f(x)=(x^2-3)^{12}[/math]. Find the coefficient of [math]x^{20}[/math].

Consider [math]\displaystyle{\left(x^2+\frac{1}{ax}\right)^{10}}[/math], find [math]a[/math] such that the coefficient of [math]x^{11}[/math] is 15.

Find the following remainders

(a) [math]6^{200}[/math] divided by 5.

(b) [math]6^{200}[/math] divided by 25.

Find [math]\binom{20}{0}-2\binom{20}{1}+4\binom{20}{2}+\cdots-2^{19}\binom{20}{19}+2^{20}\binom{20}{20}[/math].

If [math]1\times \binom{20}{1}+2\times\binom{20}{2}+\cdots+19\times \binom{20}{19}+20\times\binom{20}{20}=a\cdot2^{20}[/math], what is the value of [math]a[/math]?

(Calculator) Consider the expansion of [math](x-2y-z)^{8}[/math].

(a) How many terms are in the expansion?

(b) What is the coefficient of [math]x^3y^2z^3[/math]?

(Calculator) Find the coefficient of [math]x^2[/math] in the function [math]f(x)=(x^2-2)^6(3x+1)^{5}[/math]

(Calculator) Find the coefficient of [math]x^2[/math] in the function [math]f(x)=(x^2+2x-3)^{10}[/math].

Consider the expansion of [math](x+1)^n[/math] where [math]n[/math] is a positive integer. Suppose the coefficient of [math]x^5[/math] is the same as the coefficient of [math]x^{17}[/math]. What is [math]n[/math]?