Question

What is the coefficient of the x^5y^5- term in the biomial expansion of (2x-3y)^10

Answer 1

The coefficient of
`x^(5) * y^(5) \text { is }=\left(252 * 2^(5) *(-3)^(5)\right)=252 * 32 * 243=1959552`

Solution:

The given expression is
`(2 x-3 y)^(10)`

As per binomial theorem, we know,

`(x+y)^(n)=\sum n C_(k) x^(n-k) y^(k)`

Now here a = 2x, b = (- 3y) and n = 10 and k = 0,1,2,….10

Now
`x^(5)* y^5` will be the 6 term where k =5

Now,
`\mathrm{T}_(6)=10 \mathrm{C}_(5) *(2 \mathrm{x})^((10-5)) *(-3 \mathrm{y})^(5)=10 \mathrm{C}_(5) 2^(5) * \mathrm{x}^(5) *(-3)^(5) * \mathrm{y}^(5)`

So, the coefficient of
`x^(5) * y^(5) \text { is }=10\left(5 * 2^(5) *(-3)^(5)\right`.

`10 \mathrm{C}_(5)=(10 !)/(5 ! *(10-5) !)=(10 !)/(5 !+5 !)=(10 * 9 * 8 * 7 * 6 * 5 !)/(5 ! * 5 !)=(10 * 9 * 8 * 7 * 6)/(5 * 4 * 3 * 2 * 1)=\left((30240)/(120)\right)=252`

The coefficient of
`x^(5) * y^(5) \text { is }=\left(252 * 2^(5) *(-3)^(5)\right)=252 * 32 * 243=1959552`