What is the coefficient of the x⁹y-term in the binomial expansion of (2y + 4x³)⁴?

Solution:

The given binomial expansion is (2y + 4x³)⁴ ------------------- (1)

The general form of binomial expansion is (a + b)ⁿ -------- (2)

Comparing (1) and (2)

a = 2y 

b = 4x³

n = 4

We have to find the coefficient of the term x⁹y

This implies r = 3

The terms in the expansion can be obtained using

\(T_{r+1}=\, ^{n}C_{r}a^{(n-r)}b^{r}\)

Now,\(\\T_{3+1}=\, ^{4}C_{3}(2y)^{(4-3)}(4x^{3})^{3}\\T_{4}=\, ^{4}C_{3}(2y)^{1}(4x^{3})^{3}\\T_{4}=\, ^{4}C_{3}(2y)(4)^{3}(x)^{9}\\T_{4}=\, ^{4}C_{3}(2)(4)^{3}x^{9}y\)

\(T_{4}=\frac{4\times 3\times 2\times 1}{(3\times 2\times 1)(1)}128x^{9}y\\T_{4}=4(128)x^{9}y\\T_{4}=512x^{9}y\)

Therefore, the coefficient of the term x⁹y is 512.

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What is the coefficient of the x⁹y-term in the binomial expansion of (2y + 4x³)?

Summary:

The coefficient of the x⁹y-term in the binomial expansion of (2y + 4x³)⁴ is 512.