\problemdata      
        {?} % latex-friendly label for the prob.
        {Counting} % The topic(s) of the problem content.
        {?} % Source (if known)
        {F01, TU8; S99, PS7} % Usage (list ones you are aware of).
        {Christos Kapoutsis, S02} % Last revision info (author, date).

\begin{problem}
In each case, state whether the argument is correct or incorrect,
and explain why.  If incorrect, try to fix it.
\begin{problemparts}

\problempart Since there are 26 letters in the English alphabet, the
Product Rule implies that the number of four-letter English words is
$26^4$.

\solution{Incorrect.  Not every four letter sequence is an English word,
\eg ``xxxx'' is not English.  In fact, there are considerably fewer
four-letter English words than $26^4$.  There is no fix to this mistaken
application of the Product Rule.}

\problempart The number of 4-digit (decimal whole) numbers which have a
digit 7 somewhere is $4 * 10^3$, since by the sum rule, we must add the
counts for each position the 7 could be in; and since for each such
position, there are $10^3$ possible numbers by the product rule.  (Assume
numbers can have leading zeros)

\solution{Incorrect, since it double counts numbers containing more than
one 7 (for example 7707 is counted 3 times).  The sum rule does not apply.
Correct answer is the number of all possible 4-digit number minus the
number of those which do \emph{not} contain 7, namely, $10^4 - 9^4$.}

\end{problemparts}
\end{problem}