Solve Binomial Expansion Homework: (a) (1-x6)4, (b), (c) |x|<1

[chrisyuen]

Homework Statement

(a) Expand (1-x⁶)⁴
(b) Find the coefficient of x^r, where r is a non-negative integer, in the expansion of (1-x)^-4 for |x|<1.
(c) Using (a) and (b), or otherwise, find the coefficient of x^8 in the expansion of ((1-x⁶)/(1-x))⁴ for |x|<1.

(Answers:
(a) 1-4x⁶+6x¹²-4x¹⁸+x²⁴
(b) (r+1)(r+2)(r+3)/6
(c) 125)

Homework Equations

Binomial Expansion: n(n-1)...(n-r+1)*x^r/r!

The Attempt at a Solution

I can solve parts (a) and (b) but not part (c).

I tried (-4)(2+3)(2+2)(2+1)/6 = -40 but not correct.

Can anyone tell me how to solve it?

Thank you very much!

 

[tiny-tim]

Hi chrisyuen!

(c) Using (a) and (b), or otherwise, find the coefficient of x^8 in the expansion of ((1-x⁶)/(1-x))⁴ for |x|<1.

(Answers:
(a) 1-4x⁶+6x¹²-4x¹⁸+x²⁴
(b) (r+1)(r+2)(r+3)/6
(c) 125)
…
I tried (-4)(2+3)(2+2)(2+1)/6 = -40 but not correct.

Yes, that's x⁶ times x² …

now what about 1 times x⁸?

 

[chrisyuen]

Hi chrisyuen!


Yes, that's x⁶ times x² …

now what about 1 times x⁸?

Yes, I got it!

Thank you very much!