By the binomial theorem,
[tex]\displaystyle (2x+y)^7 = \sum_{k=0}^7 \binom7k (2x)^{7-k} y^k=\sum_{k=0}^7 \binom7k 2^{7-k} x^{7-k} y^k[/tex]
where
[tex]\dbinom nk=\dfrac{n!}{k!(n-k)!}[/tex]
is the binomial coefficient.
We get the x⁴y³ terms with k = 3, for which the coefficient would be
[tex]\dbinom73 2^{7-3}=\dfrac{7!}{3! 4!} 2^4=\boxed{560}[/tex]
