Using the binomial theorem to expand (1+4x)^4​

!

Explanation:

The coefficient of the kth term (ordering in increasing order for the exponent of x) is just the number of ways we have to choose k factors from that expression, so if we let k be the exponent of x, and n be the total number of terms, the coefficient of

is

. For example, the coefficient of the third grade te

W₁=0.15 (15%)
p₁=1.0582 g/ml=1058.2 g/L
v₁-?
w₂=0.35 (35%)
p₂=1.1482 g/ml=1148.2 g/L
v₂-?
w₃=0.19 (19%)
v₃=35L

v₁+v₂=v₃  (1)

w₁=m₁/v₁p₁   m₁=v₁p₁w₁
w₂=m₂/v₂p₂   m₂=v₂p₂w₂

w₃=(v₁p₁w₁+v₂p₂w₂)/(v₁p₁+v₂p₂)
v₁p₁(w₃-w₁)=v₂p₂(w₂-w₃)
v₁/v₂=p₂(w₂-w₃)/[p₁(w₃-w₁)]  (2)

We solve the equations (1) and (2).
v₁+v₂=35
v₁/v₂=1148.2*0.16/[1058.2*0.04]=4.34

v₁=28.446 L
v₂=6.554 L