Question

What is the coefficient of x^2y^3 in the expansion of (2x+y)^5

Answer 1

We figure out the coefficient of x^2y^3 by figuring out the coefficients in both (x+y)^4 and (x+2y)^4. 

Using the binomial theorem, the coefficient of x^2y^3 on the first expansion is 

(4 choose 3)x^2y^3=6x^2y^3, 

and the coefficient of x^2y^3 on the second expansion is 

(4 choose 3)x^2(2y)^3=6x^2(4y^2)=24x^2y^3. 

Thus, the coefficient in the sum of these two expansions is 6+24=30.