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Add solution to exercise 1.2.6 31.

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1 parent f5cca8e commit 4a6015410cbdd8de7cc746e2e39ec77558a5d5f1 @l0stman committed Feb 7, 2012
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  1. +21 −0 chap1/1.2/1.2.6/1.2.6.tex
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21 chap1/1.2/1.2.6/1.2.6.tex
@@ -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}

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