binomial theorem and expansions

Question

can anyone please explain these things:

formula for (1-x)ⁿ

formula for 1/(1-x)ⁿ

general term in expansion of (1-x)ⁿ and 1/(1-x)ⁿ

and coeffecient of a term in these expansions.

please elaborate a little because i have read few questions on generating functions and binomial where these things are used but i am getting very confused.i dun know much about them and gathering info from internet is also confusing me.

Comments

Kenneth Rosen Adv counting chapter

Answer 1

$(1-x)^{n}$=$_{0}^{n}\textrm{c}x^{0}-_{1}^{n}\textrm{c}x^{1}+_{2}^{n}\textrm{c}x^{2}-......$
in this signs are alternative.

co-efficeint of $x^{r}=_{r}^{n}\textrm{c} (-1)^{r}$

$(1-x)^{-n}=_{0}^{n}\textrm{c}x^{0}+_{1}^{n}\textrm{c}x^{1}+_{2}^{n}\textrm{c}x^{2}+......$
in this all signs are positive.

co-efficeint of $x^{r}=_{r}^{n}\textrm{c}$

if expression is in the form of $(x-y)^{n}$ then common 'x' term  we get $x^{n}(1-y/x)^{n}$

again apply above formula

Comments

thanks a lot :-)