- #1

gruba

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## Homework Statement

Find [itex]\sum\limits_{k=0}^{n}k^2{n\choose k}(\frac{1}{3})^k(\frac{2}{3})^{n-k}[/itex]

## Homework Equations

-Binomial theorem

## The Attempt at a Solution

I am using the binomial coefficient identity [itex]{n\choose k}=\frac{n}{k}{{n-1}\choose {k-1}}[/itex]:

[tex]\sum\limits_{k=0}^{n}k^2{n\choose k}(\frac{1}{3})^k(\frac{2}{3})^{n-k}=\sum\limits_{k=1}^{n-1}\frac{n}{k}{{n-1}\choose {k-1}}(k-1)^2(\frac{1}{3})^{k-1}(\frac{2}{3})^{n-k+2}[/tex]

What am I doing wrong here (sums are not equal)?