MATHEMATICS  PROBLEM  #121

     The Maryland lottery "Big Game" is played by selecting five of the
numbers 1, 2, 3, . , 50 and then selecting one of the numbers 1, 2, 3,..., 36.
The first selection must consist of five different numbers with no repeats,
and the order in which they are picked does not matter.  The second selection
may or may not repeat one of the first five numbers.
     How many ways can this be done?  Put differently, how many plays would
you have to purchase (for one dollar each) in order to be sure that you have
the winning play?  (Be sure to include a careful explanation of your
reasoning and calculations in your solution!)


      Each midshipman submitting a correct solution with a correct
explanation to Problem 121 by 1700 on Thursday 31 January 2002 will win a 
cookie.  Submit solutions to Prof. Wardlaw at mathprob@usna.edu (please no 
attachments!) or via his mailbox in Chauvenet 301.

      A correct solution to Mathematics Problem #120 was submitted by
Midshipman Alan M. Montara.  Professors Gary Fowler, Mark Kidwell, and Mark
Meyerson also submitted solutions to Problem #120.  My solution to Problem 
#120 is posted on the board and is given below.


 MATHEMATICS  PROBLEM  #120

     Midshipman Joe asks Midshipman Bob "Is the following expression
correct?" and writes

                     f '(x^3) = 3x^2

on the blackboard.  Bob replies, "Well it could be, but I don't think
that's what you mean."

      What do you think Joe meant?  Find a function which makes what Joe
wrote correct.

 Solution.  Joe probably thought that  " f ' " meant  "take the derivative
of what follows", or  f ' = d/dx.
     To find the function  f   which makes Joe's statement  f '(x^3) = 3x^2
correct, we let   x^3 = t,  so that   t = x^(1/3)  and substitute to get   
f'(t) = f '(x^3) = 3x^2  =  3t^(2/3).  It follows that   f '(x)  =  3x^(2/3);  
integration shows that    f(x) =  (9/5)x^(5/3) + c is a function which makes 
Joe's statement correct.