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

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.