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\date{Spring 2016}


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{\bf Math 432 -- Combinatorics\\
Homework 1}\\
Due  Friday Jan. 29 midnight, 2016.
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Solve the following problems. Show all your work. Please type your homework by way of Latex and submit your .pdf file through blackboard.   Each problem is 4 points if not specified.



\begin{enumerate}
\item (6 points) In terms of binomial coefficients, count the (5-card) poker hands that have \\
a) three-of-a-kind  (three cards of the same rank and one in each of two other ranks).\\
b) full house (two cards of equal rank and three cards of another rank).\\
c) straight flush (five cards in sequence from the same suite).  




\item Prove the following identity
\[  \sum_{k=0}^n k{n\choose k}=n 2^{n-1}. \]


\item Prove the following identity:
\[
\sum_{i=0}^k {m\choose i}{n\choose k-i}={m+n\choose k}.
\]


\item By calculating the coefficient of $x^n$ on the two sides of the identity $(1+x)^n\cdot (1-x)^n=(1-x^2)^n$, prove that 
\[
\sum_{k=0}^n (-1)^k {n\choose k}^2=\begin{cases} 0, \text{ if $n$ is odd}\\  (-1)^m{2m\choose m}, \text{if $n=2m$.}\end{cases}
\]

\item Compute $\sum_{k=0}^n k^2{n\choose k}$. 



\end{enumerate}

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