1. This is just a case of allowing Maple to evaluate the sums, and then
simplifying them.
a) [Hint: to simplify this you must use first "convert(%,factorial)",
then "simplify(%)", by eye it is easy to see the cancellation which
occurs among the 4 factors in factorial form.]
b) [Here simplify is enough, since the cancellation occurs outside the binomial
coefficient factor, and is trivial to see: the last term in the numerator is -2
times the denominator.]
2. Fill in the discussion from the example in the section 4.2 file so that it makes a complete report. Don't just type in the code in the back of the book, but try to set up why that then finishes off the problem.
5. This is interesting and more elementary than some of the others, following up on a problem already begun in a preliminary worksheet.
6. This is the 2d random walk problem:
http://en.wikipedia.org/wiki/Random_walk .
Understand and explain the code given in the back of the book.
7. This is straightforward following the similar example in the chapter, using linear algebra to find the coefficients in the 2nd order recursion relation.
8. Again relatively straightforward following up examples with rand() in a preliminary worksheet.
9. This is really interesting. See also:
http://en.wikipedia.org/wiki/Benfords_law
Remember, the evenness or oddness of the product X Y Z is an asymmetric
distribution because of the nonrandomness built into products of even and odd
numbers. Here too the leading digit of products of numbers has nonrandomness built into it.
This problem experimentally generates an approximation to this distribution.
Compare with the webpage explanation afterwards. Can one deduce from it, the
theoretical prediction for the leading digit of the product? [I don't know
myself.]
Google:
http://www.google.com/search?hl=en&q=benford's+law+multiplying+numbers&aq=f&oq=