There are four types of common coins in   US   currency:

   quarters   (25 cents)
   dimes   (10 cents)
   nickels   (5 cents),   and 
   pennies   (1 cent) 


There are six ways to make change for 15 cents:

   A dime and a nickel 
   A dime and 5 pennies
   3 nickels
   2 nickels and 5 pennies
   A nickel and 10 pennies
   15 pennies


Task

How many ways are there to make change for a dollar using these common coins?     (1 dollar = 100 cents).



Optional

Less common are dollar coins (100 cents);   and very rare are half dollars (50 cents).   With the addition of these two coins, how many ways are there to make change for $1000?

(Note:   the answer is larger than   232).



References

 an algorithm from the book Structure and Interpretation of Computer Programs.
 an article in the algorithmist.
 Change-making problem on Wikipedia.


function countSums (amount, values)
    local t = {}
    for i = 1, amount do t[i] = 0 end
    t[0] = 1
    for k, val in pairs(values) do
        for i = val, amount do t[i] = t[i] + t[i - val] end
    end
    return t[amount]
end
 
print(countSums(100, {1, 5, 10, 25}))
print(countSums(100000, {1, 5, 10, 25, 50, 100}))