\problemdata                                                             
        {Increasing Sequences} % latex-friendly label for the prob.
        {Counting} % The topic(s) of the problem content.  
        {?} % Source (if known)                                          
        {S01 Tutorial 9, 1} % Usage (list ones you are aware of). 
        {Tina Wang, S02} % Last revision info (author, date).   
                                                                         
                                                                         
\begin{problem}           
\begin{problemparts}
\problempart How many increasing sequences are there of $k$ numbers from the range
$(1 \dots n)$?

\solution{
Since every subset of the range gives a unique increasing sequence, the problem
is equivalent to simply selecting $k$ numbers from $n$: $\binom{n}{k}$.
} 

\problempart How many increasing subsequences of any length are there that take
numbers from $(1 \dots n)$? 

\solution{
Using our solution to the previous part, this is the sum of all binomial
coefficients for $n$:
\[\sum_{k=0}^n \binom{n}{k} = 2^n.\]
}

\end{problemparts}
\end{problem}                      