SOLUTION: You have 3 quarters, 8 dimes, 5 nickels, 7 pennies how many combinations add up to $1 & list all combinations.

1. If you don't use any quarters, you have to use 7 or 8 dimes, because if you
used only 6, you'd have only 60 cents, and you couldn't make the remaining 40
cents with only 5 nickels and 7 pennies.

If you use 7 dimes, you have 30 cents to make, and you can only do that with 
5 nickels and 5 pennies.   That's 1 way.

If you use 8 dimes, you have 20 cents to make, and you can do that with
either 4 nickels, or 3 nickels and 5 pennies.   That's 2 ways.

That's 1+2 or 3 ways to not use any quarters.

---------------------

2. If you use 1 quarter, you have 75 cents left. So you have to use 5, 6 or 7
dimes, because if you used only 4, you'd have only 40 cents, and you couldn't
make the remaining 35 cents with only 5 nickels and 7 pennies, for that's only

32 cents.

If you use 5 dimes, you have 25 cents to make, and you can make that with
either 5 nickels or with 4 nickels and 5 pennies.   That's 2 ways.

If you use 6 dimes, you have 15 cents to make, and you can make that with
3 nickels or with 2 nickels and 5 pennies.   That's 2 ways.

If you use 7 dimes, you have 5 cents to make, and you can make that with
1 nickels or 5 pennies.   That's 2 ways.

So that's 2+2+2 or 6 ways to use exactly 1 quarter.

---------------------------------

3, If you use 2 quarters, you have 50 cents left. So you have to use 2, 3, 4 or
5 dimes, because if you used only 1, you'd have only 10 cents, and you couldn't
make the remaining 40 cents with only 5 nickels and 7 pennies, for that's only
32 cents.

If you use 2 dimes, you have 30 cents to make, and you can only make that with
either 5 nickels and 5 pennies.   That's 1 way.

If you use 3 dimes, you have 20 cents to make, and you can make that with 

4 nickels or with 3 nickels and 5 pennies.   That's 2 ways.
If you use 4 dimes, you have 10 cents to make, and you can make that with
2 nickels or 1 nickel and 5 pennies.   That's 2 ways.

If you use 5 dimes, then that the whole 50 cents.  So that's 1 way

So that's 1+2+2+1 = 6 ways to use exactly 2 quarters.

---------------------------------

4. If you use 3 quarters, you have 25 cents left. So you don't have to use any
dimes or you can use 1 or two dimes. 

If you don't use any dimes, then you can use 5 nickels, or 4 nickels and 5
pennies.  That's 2 ways.

If you use 1 dime, you have 15 cents to make, and you can make that with either
3 nickels or 2 nickels and 5 pennies.   That's 2 ways.

If you use 2 dimes, you have 5 cents to make, and you can make that with
1 nickel or 5 pennies.   That's 2 ways.

So that's 2+2+2 or 6 ways to use exactly 3 quarters.

---------------------------------

So the total number of ways to have $1 is 3+6+6+6 = 21 ways.

Edwin