let

x = the number of 5-peso coins, x is integer, x>0

y = the number of 20-peso bills, y is integer, y>0

5x + 20y = 500 divide both sides by 5

x + 4y = 100

x = 4(25 - y)

this means x can be written as 4k, where k is integer, k>0

4k + 4y = 100 divide both sides by 4

k + y = 25

y = 25 - k

for k an element of {1, 2, 3,....,24} we get the solutions for x = 4k and y = 25 - k

since {1, 2, 3,....,24} has 24 elements thus the answer is in 24 ways,