# Binomial questions 2

• Dec 19th 2012, 09:25 PM
Tutu
Binomial questions 2
Hi I've some more questions, your help is greatly appreciated!

1.) $\displaystyle (x^2-3x+1)^2=x^4-gx^3+11x^2-6x+1$ and the sum of it's coefficients is 1, what is the sum of the coefficients of
$\displaystyle (x^3+2x^2+3x-7)^{100}?$

I really have no idea how to start, can't see any patterns..

2a.) Prove that $\displaystyle r{n \choose r}=n{n-1 \choose r-1}$

I did this ok so ignore this part! I had just typed it down as this part may yield a clue for the other parts.

2b.) Hence show that $\displaystyle {n \choose 1}+2{n \choose 2}+3{n \choose 3}+4{n \choose 4}+...+n{n \choose n}=n2^{n-1}$

2c.) Suppose the numbers Pr ( r is a subscript!) are defined by
$\displaystyle Pr={n \choose r} p^r(1-p)^{n-r} for r=0,1,2,3...,n.$

Prove that

$\displaystyle \sum\limits_{r=0}^nPr=1$

and that

$\displaystyle \sum\limits_{r=1}^nrPr=np$

Sorry for question 1, it is not $\displaystyle gx^3$ but $\displaystyle 6x^3$, sorry for the typo and thank you!