@ is said to obey associative law iff a @ (b @ c) = (a @ b) @ c.
@ is said to obey distributive law over # iff a @ (b # c) = (a @ b) # (a @ c)
A. $a \diamond (b \diamond c) = a + 2b + 4c$
$(a \diamond b) \diamond c = a + 2b + 2c$
$\implies a \diamond (b \diamond c) \neq (a \diamond b) \diamond c$
B. $a \square (b \square c) = abc$
$(a \square b) \square c = abc$
$\implies a \square (b \square c) = (a \square b) \square c$
C. $a \diamond (b \square c) = a + 2bc$
$(a \diamond b) \square (a \diamond c) = (a+2b)(a+2c)$
$\implies a \diamond (b \square c) \neq (a \diamond b) \square (a \diamond c)$
D. $a \square (b \diamond c) = a(b+2c) = ab + 2ac$
$(a \square b) \diamond (a \square c) = ab + 2ac$
$\implies a \square (b \diamond c) = (a \square b) \diamond (a \square c)$
Answer - B, D