MATHEMATICS  PROBLEM  #120
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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.

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Each midshipman submitting a correct solution with a correct explanation to
Problem 120 by 1700 on Friday 30 November 2001 will win a cookie.  Submit
solutions to Prof. Wardlaw at mathprob@usna.edu (please no attachments!) or
via his mailbox in Chauvenet 301.
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No correct solutions to Mathematics Problem #119 were submitted by
midshipmen.  Professors Jodi Lockhart, Mark Kidwell, and Mark Meyerson
submitted solutions to Problem #119.  My solution to Problem #119 is posted
on the board and below.

MATHEMATICS  PROBLEM  #119

A spider puts on 8 identical socks and 8 identical shoes.  In how many
orders can the spider do this, given that on each foot, the sock has to go
on before the shoe?

 Solution.  There are  (16!)/(28) = 81,729,648,000  different orders that
the spider can put on the shoes and socks.  This is because there are  16!
orders to accomplish the  16  tasks, but this number must be divided by  28
to account for the sock having to go on before the shoe on each foot.
Perhaps this can be made clearer as follows:  The spider needs to decide
when, among the 16 time slots available, to put the sock on each foot and
when to put the shoe on each foot.  This amounts to choosing  two different
numbers between  1  and  16  for each foot, the smaller to designate when
the sock goes on that foot and the larger to designate when the shoe goes on
that foot.   We can count the number of ways of doing this by counting the
number of ways that  16  pool balls, numbered  1, 2, 3, ., 15, 16  can be
put into eight numbered bags, with two balls going into each bag.  Filling
the bags in order, there are  16  choices for the first ball to go into the
first bag, and  15  choices for the second ball, but we divide by  2
because we would get the same result if we reversed the order in which these
balls were picked.  Thus there are  16x15/2  different ways to fill the
first bag.  Now there are 14  balls left over, so there are  14x13/2  ways
to fill the second bag.  Similarly, there are 12x11/2 ways to fill the third
bag,  10x9 /2  ways to fill the fourth bag, etc., so there are

(16x15/2)(14x13/2)(12x11/2)...(4x3/2)(2x1/2)
= (16x15x14x13x12x11x10x9x8x7x6x5x4x3x2x1)/(2x2x2x2x2x2x2x2)
=  (16!)/(28) = 81,729,648,000

different orders in which the socks and shoes can be put on.