I am trying to solve a combinatorics problems where I need to compute a sum based on binomial coffecients:

$${\sum_k}{\frac{1}{k+1}}{99\choose k}{200\choose 120 - k} $$

The only pattern that I can see here is :

$${\sum_k}{99\choose k}{200\choose 120 - k} = {99 + 200\choose 120}$$

My guess has been that I need to incorporate the $${\frac{1}{k+1}}$$ in order to match the form:

$${\sum_k}{n\choose k}{m\choose l - k} = {m + n\choose l}$$

I have tried decomposing the binomial coefficients into their factorial form:

$${\sum_k}{\frac{1}{k+1}}{99\choose k}{200\choose 120 - k} $$ = $${\sum_k}{\frac{1}{k+1}}{\frac{99!}{k! * (99 - k)!}}{\frac{200!}{(120 - k)! * (200 - (120 - k))!}} $$


$${\sum_k}{\frac{99!}{(k + 1)! * (99 - k)!}}{\frac{200!}{(120 - k)! * (200 - (120 - k))!}} $$

I do not see any patterns here that can lead me to the correct answer. Any advice or direction would be appreciated! Please do not post a solution!


You may get rid of that $\frac{1}{1+k}$ in the following way: $$ \frac{1}{k+1}\binom{99}{k} = \frac{99!}{(k+1)!(99-k)!}=\frac{1}{100}\binom{100}{k+1} $$ then exploit a well-known convolution. You are essentially computing the coefficient of $x^{121}$ in the product between $(1+x)^{100}$ and $(1+x)^{200}$.

  • 1
    $\begingroup$ Thanks! I applied Vandermonde's convolution to the result, if anyone was interested. $\endgroup$ – jonnywalkerr Oct 1 '15 at 19:38

One more trick: $\frac{1}{k+1} = \int_{0}^{1} x^k dx$ to get rid of the fraction.


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