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I have to evaluate the following but I am not sure how to do it. My professor went over it briefly trying to get it in before the end of the semester but did not explain it very well. I'm not even sure of where to begin.

$$ \sum\limits_{k=7}^{60}\binom{k-2}{20}. $$

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Welcome to math.stackexchange! If you want your latex expression to render as a mathematical expression and not as a text, you can put it between "\$\$...\$\$". –  levap Dec 16 '12 at 1:41
    
Thank you, I was trying to figure out how to do that! –  user51754 Dec 16 '12 at 1:43
    
Also, are you sure your expression is correct? If you start at $k = 7$, the expression at the top of the binomial coefficient is smaller than the expression at the bottom. –  levap Dec 16 '12 at 1:44

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The binomial identity $$ \sum_{k=i}^{n-j}\binom{k}{i}\binom{n-k}{j}=\binom{n+1}{j+1} $$ setting $j=0$, says that $$ \sum_{k=20}^{58}\binom{k}{20}=\binom{59}{21} $$ which is your sum after ignoring terms that are $0$.

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