How many combinations are there to make forty cents and what are they?

Just start with 1 quarter, 1 dime, 1 nickel. Then gradually swap out the lower coins with pennies. Once you have exhausted the smaller coins, take away a larger coin and repeat the process:

There are 31 ways to make 40 cents

1Q, 1D, 1N

1Q, 1D, 5P

1Q, 3N

1Q, 2N, 5P

1Q, 1N, 10P

1Q, 15P

4D

3D, 2N

3D, 1N, 5P

3D, 10P

2D, 4N

2D, 3N, 5P

2D, 2N, 10P

2D, 1N, 15P

2D, 20P

1D, 6N

1D, 5N, 5P

1D, 4N, 10P

1D, 3N, 15P

1D, 2N, 20P

1D, 1N, 25P

1D, 30P

8N

7N, 5P

6N, 10P

5N, 15P

4N, 20P

3N, 25P

2N, 30P

1N, 35P

40P

P.S. King Lucario had fewer combinations but then decided to copy my solution. Please award best answer to a person that actually came up with the combinations on his own.

40 pennies

35 pennies, 1 nickel

30 pennies, 2 nickels

30 pennies, 1 dime

25 pennies, 3 nickels

25 pennies, 1 nickel, 1 dime

20 pennies, 4 nickels

20 pennies, 2 nickels, 1 dime

20 pennies, 2 dimes

15 pennies, 5 nickels

15 pennies, 3 nickels, 1 dime

15 pennies, 1 nickel, 2 dimes

15 pennies, 1 quarter

10 pennies, 6 nickels

10 pennies, 4 nickels, 1 dime

10 pennies, 2 nickels, 2 dimes

10 pennies, 1 nickel, 1 quarter

10 pennies, 3 dimes

5 pennies, 7 nickels

5 pennies, 5 nickels, 1 dime

5 pennies, 3 nickels, 2 dimes

5 pennies, 2 nickels, 1 quarter

5 pennies, 1 nickel, 3 dimes

5 pennies, 1 dime, 1 quarter

8 nickels

6 nickels, 1 dime

4 nickels, 2 dimes

3 nickels, 1 quarter

2 nickels, 3 dimes

1 nickel, 1 dime, 1 quarter

4 dimes

31 combinations

1Q, 1D, 1N

1Q, 1D, 5P

1Q, 3N

1Q, 2N, 5P

1Q, 1N, 10P

1Q, 15P

4D

3D, 2N

3D, 1N, 5P

3D, 10P

2D, 4N

2D, 3N, 5P

2D, 2N, 10P

2D, 1N, 15P

2D, 20P

1D, 6N

1D, 5N, 5P

1D, 4N, 10P

1D, 3N, 15P

1D, 2N, 20P

1D, 1N, 25P

1D, 30P

8N

7N, 5P

6N, 10P

5N, 15P

4N, 20P

3N, 25P

2N, 30P

1N, 35P

40P

N = Nickels

P = Pennies

D = Dimes

Q = Quarters