SOLUTION: How many ways can a person with four nickels, three dimes, and 2 quarters make a $.65 purchase from a machine?

Notice that 4 nickels + 3 dimes = 4*5 + 3*10 = 50 cents.


It means, that one or two quarters must participate.



If two quarters participate, then the remaining 15 cents can be comprised of 

   EITHER  1 nickel and 1 dime 

   OR      3 nickels, 

giving 2 possible combinations.



If only one quarter does participate, then the remaining 65-25 = 40 cents 
can be comprised of

   EITHER  4 nickels and 2 dimes,

   OR      2 nickels and 3 dimes,

giving another 2 possible combinations.



So, in all, there are 2 + 2 = 4 possible combinations to make a 65 cents purchase WITHOUT HAVING a change.