Question

Expand the binomial (3x^2 + 2y^3)^4.

The coefficient of the third term in the expansion of the binomial (3x2 + 2y3)4 is

Answer 1

To expand a binomial of the form $(a+b)^n$, we are going to use Pascal's triangle.
We can infer from our problem that $a=3x^2$, $b=2y^3$, and $n=4$. Since $n=4$, we are going to use the fourth row of Pascal's triangle to get the coefficients of our expansion, so the coefficients of our expansion will be: 1      4     6     4     1
Now that we have our coefficients, we cant expand our binomial:
$(3x^2+2y^3)^4=1(3x^2)^4(2y^3)^0+4(3x^2)^3(2y^3)^1+6(3x^2)^2(2y^3)^2+$$4(3x^2)^1(2y^3)^3+1(3x^2)^0(2y^3)^4$

$=81x^8+216x^6y^3+216x^4y^6+96x^2y^9+16y^2$

Since the third term of the expansion is $216x^4y^6$, we can conclude that the coefficient of the third term is 216.