SOLUTION: Use a generating function to model the problem: How many ways are there to make change for $0.62 using only pennies, nickels, and dimes so that the total number of coins used is ev

1.  On the method of generating function/functions to solve coin problems see the pretty fine explanation at the link

    http://www.cs.cmu.edu/afs/cs.cmu.edu/academic/class/15251-f10/Site/Materials/Lectures/Lecture09/lecture09.pdf

    http://www.cs.cmu.edu/afs/cs.cmu.edu/academic/class/15251-f10/Site/Materials/Lectures/Lecture09/lecture09.pdf



2.  But I think that the GIVEN particular problem can be solved faster and easier DIRECTLY and EXPLICITLY:


   2 pennies  +  0 dimes   + 12 nickels
   2 pennies  +  2 dimes   +  8 nickels
   2 pennies  +  4 dimes   +  4 nickels
   2 pennies  +  6 dimes   +  0 nickels

   7 pennies  +  0 dimes   + 11 nickels
   7 pennies  +  2 dimes   +  7 nickels
   7 pennies  +  4 dimes   +  3 nickels

  12 pennies  +  0 dimes   + 10 nickels
  12 pennies  +  2 dimes   +  6 nickels
  12 pennies  +  4 dimes   +  2 nickels

  17 pennies  +  0 dimes   +  9 nickels
  17 pennies  +  2 dimes   +  5 nickels
  17 pennies  +  4 dimes   +  1 nickels

  22 pennies  +  0 dimes   +  8 nickels
  22 pennies  +  2 dimes   +  4 nickels
  22 pennies  +  4 dimes   +  0 nickels

  27 pennies  +  0 dimes   +  7 nickels
  27 pennies  +  2 dimes   +  3 nickels

  32 pennies  +  0 dimes   +  6 nickels
  32 pennies  +  2 dimes   +  2 nickels

  37 pennies  +  0 dimes   +  5 nickels
  37 pennies  +  2 dimes   +  1 nickels

  42 pennies  +  0 dimes   +  4 nickels
  42 pennies  +  2 dimes   +  0 nickels

  47 pennies  +  0 dimes   +  3 nickels

  52 pennies  +  0 dimes   +  2 nickels

  57 pennies  +  0 dimes   +  1 nickels

  62 pennies  +  0 dimes   +  0 nickels


In this way I counted 28 possibilities.


So, my answer is:  28 possibilities.