\kproblem\ksource{...}
% induction

\kproblem\ksource{...}
% tournaments/ranking

\kproblem\ksource{Nancy/Carl/Victor}
% injections/surjections and counting 

Which of the following is the cardinality of the set of all
surjections from a set $A$, where $|A|=n$, and $B$, where $|B|=m$.
\begin{description}\itemsep 0\baselineskip
\item[(a)] $m^n$
\item[(b)] $\frac{m!}{(m-n)!}$
\item[(c)] ${n \choose m} m^{n-m} m!$
\item[(d)] None of the above.
\end{description}

\solution{%
  (d) None of the above.}

\kproblem\ksource{...}
% pigeonhole

\kproblem\ksource{Nancy/Carl}

Suppose you have a collection of 41 balls, of which you know that 40
have normal weight and one is abnormal~---~either heavy or light, but
you don't know which.

Is there a way to determine which ball is abnormal with 4 weighings?
If so, demonstrate the approach, otherwise prove that it is
impossible.

\solution{%
  No.  The 4 weighings can only generate 81 distinct outcomes.
  However, any of the 41 balls can be the abnormal one and the
  abnormal one can be either light or heavy, i.e., 82 distinct
  outcomes.}

\kproblem\ksource{Carl...}
% recurrences and summations... simple!

\ppart Solve the following homogeneous recurrence, assuming that
$T(0)=1$ and $T(1)=1$.

$$T(n) = T(n-1) - \frac{1}{4} T(n-2)$$

\solution{
  I think the answer here is:

  $$T(n) = (1/2)^n + n \cdot (1/2)^n$$
}

\ppart Find a closed-form solution for the infinite sum:

$$\sum_{n = 0 }^{\infty} T(n)$$

\solution{
  we just sum the terms and get 4.
}
