Build a Generating Function to find the number of ways to make change for $5 using Pennies (v), Nickels (w),Dimes (x),Quarters (y),

and 1/2 Dollars (z) using a MAXIMUM of 100 coins and using at least 1 of each coin in each case

v + w + x + y + z <= 100 ......(E1)

v+1 + 5(w+1) + 10(x+1)x + 25(y+1) + 50(z+1)z = 500

v + 5w + 10x + 25y + 50z = 409 .....(E2) Sum of coin denominations = 91 , 500-91-100=309

and (E2) - (E1) gives 4w + 9x + 24y + 49z = 309

We then require the coefficient of x^309 in the expansion of

1
(E3) ----------------------------- but I think this only gives the 100 coin case.
(1-x^4)(1-x^9)(1-x^24)(1-x^49)