Why Does the Binomial Theorem Solution Differ from the Book's Answer?

[iSplicer]

Thanks, this ended up helping a lot =)

and sorry for digging up an old thread!

 

[iSplicer]

This is what I've been able to work out :

$$\sum_{r=0}^{n} r C^n_r x^r y^{n-r}$$ <--------(given)

$$= \frac{n!}{(r-1)!(n-1-(r-1)!} x^r y^{n-r}$$ <---------(cancelling the r's & adding & subtracting 1 & regrouping them)

$$= \frac{(nx)}{x} \frac{(n-1)!}{(r-1)!(n-1-(r-1)!} x^r y^{n-r}$$ <-------(separating n & multiplying & dividing by x)

$$= (nx) C^{n-1}_{r-1} x^{r-1}y^{n-r}$$ <-------( The R.H.S)

$$= nx(y+x)^{n-1}$$ <--------(The L.H.S)

$$= nx$$ <--------(Substuting the value x+y=1)


I think something's not right above, but what is it ?

Hmm, just a question: The solution seems impeccable, but what happened to the sigma notation after the first line?