# l0stman/taocp

Add solution to exercise 1.2.6 31.

 @@ -562,4 +562,25 @@ $\sum_k {n+k \choose m+2k} {2k \choose k} \frac{(-1)^k}{k+1} = {n-1 \choose m-1}.$ +\newpar{31} We have +\begin{eqnarray*} + && \sum_k {m-r+s \choose k} {n+r-s \choose n-k} {r+k \choose + m+n}\\ + &=& \sum_k \sum_j {m-r+s \choose k} {n+r-s \choose n-k}{r + \choose m+n -j}{k \choose j} \\ + &=& \sum_k \sum_j {m-r+s \choose + j}{m-r+s-j \choose k-j} {n+r-s \choose n-k} {r \choose m+n -j} \\ + &=& \sum_k \sum_j {m-r+s \choose j} {m-r+s -j \choose m-(k+r-s)}{n+r-s + \choose k+r-s} {r \choose m+n - j} \\ + &=& \sum_k \sum_j {m-r+s \choose j} {m-r+s - j \choose m-k} {n+r-s + \choose k} {r \choose m+n -j } \\ + &=& \sum_j {m-r+s \choose j} {r \choose m+n -j} \sum_k {m-r+s -j + \choose m-k} {n+r-s \choose k} \\ + &=& \sum_j {m-r+s \choose j} {r \choose m+n - j} {m + n - j \choose + m} \\ + &=& \sum_j {m-r + s \choose j} {r \choose m} {r -m \choose n - j} \\ + &=& {r \choose m} \sum_j {m-r+s \choose j} {r-m \choose n-j} \\ + &=& {r \choose m} {s \choose n} +\end{eqnarray*} + \end{document}