What is the coefficient of the x⁵y⁵-term in the binomial expansion of (2x - 3y)¹⁰?

Solution:

The given binomial expansion is (2x - 3y)¹⁰ --- (1)

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

Comparing (1) and (2)

a = 2x, b = -3y, n = 10

We have to find the coefficient of the term x⁵y⁵

This implies r = 5

The terms in the expansion can be obtained using

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

Now, \(\\T_{5+1}=\, ^{10}C_{5}(2x)^{(10-5)}(-3y)^{5}\\T_{6}=\, ^{10}C_{5}(2x)^{5}(-3y)^{5}\\T_{6}=\, ^{10}C_{5}(2)^{5}x^{5}(-3)^{5}(y)^{5}\\T_{6}=\, ^{10}C_{5}(2)^{5}(-3)^{5}x^{5}y^{5}\)

Therefore, the coefficient of the term \(x^{5}y^{5}=\, ^{10}C_{5}(2)^{5}(-3)^{5}\).