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\begin{center}
{\Huge 21-128 and 15-151 problem sheet 7}

Solutions to the following five exercises and optional bonus problem are to be submitted through 
blackboard by 8:30AM on

\textbf{Thursday 3rd November 2016.}

\end{center}

\subsubsection*{Problem 1}
How many ways are there to pick two cards from a standard $52$-card deck, such that the first card is a spade and the 
second card is not an ace?

\subsubsection*{Problem 2}
Count the number of hands of six cards from a standard deck of $52$ cards that contain at least one card of every suit.

\subsubsection*{Problem 3}
Find the number of functions $f: [6] \to [6]$ such that $f$ contains exactly three elements in its image.

\subsubsection*{Problem 4}
We roll a fair six-sided die exactly four times. For $k \in \{ 0, 1, 2, 3, 4 \}$, determine the probability that we roll 
a six exactly $k$ times. Check your answer by verifying that these probabilities sum to one.

\subsubsection*{Problem 5}
By counting in two ways, prove that $\sum_{k=1}^n 2^{k-1} = 2^n - 1$ for all $n \in \mathbb{N}$.

\subsubsection*{Bonus Problem - (2 points)} Tram tickets have six-digit numbers (from $000000$ to $999999$). A ticket 
is called \textit{lucky} if the sum of its first three digits is equal to the sum of its last three digits. 
A ticket is called \textit{medium} if the sum of all its digits is 27. Let $A$ and $B$ denote the numbers of lucky 
tickets and medium tickets respectively. Prove that $A=B$. 

\end{document}
