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If your currency has coins (or bills) in the following
denominations (100, 50, 25, 10, 5, 1), with what numbers
of them can you make a total of 100 (cents or dollars or 'units')?

Using standard US coins, here are the results:

For fewer than 10 coins, we can do it as follows:

(dollars, halves, quarters, dimes, nickels)
1: (1,0,0,0,0)
2: (0,2,0,0,0)
3: (0,1,2,0,0)
4: (0,0,4,0,0)
5: (0,1,1,2,1)
6: (0,1,0,5,0)
7: (0,0,3,1,3)
8: (0,0,3,0,5)
9: (0,0,1,7,1)
 Use this calculator
to find more combinations:
Dollars:
Halves:
Quarters:
Dimes:
Nickels:
Pennies:
Total coins:
Total value:

For more than 9 coins, there are often many ways for
a particular number. For example, for 10:
10 · 10 = 100 and
3 · 25 + 2 · 10 + 5 · 1 = 100 and
2 · 25 + 2 · 10 + 6 · 5 = 100

But using just dimes, nickels, and pennies we have:

The number of pennies has to be a multiple of 5,
because all the other coins are multiples of 5, and so is 100.

For '5 x p' pennies, for p from 0 to 14,
(100 - 5p)/5 = 20 - p nickels can be added,
to make a total of total of 20 + 4p coins.

So we are covered for any multiple of 4 from 20 to 76.
But, since each of those uses at least 6 nickels,
we can change 2 nickels into 1 dime which uses 1 coin less,
and also use 20 + 4p - (1,2,3,4) coins, which covers
every number from 17 up to 76.

Since 20 nickels and 0 pennies make 100, we can do this
"1 dime = 2 nickels" trick even further back from 20,
all the way to 10 dimes = 100.

So we are covered now from 1 to 76.

At 77 we run into trouble:

If we use 75 pennies, we only have 2 coins left to make the
other 25. We can't use a quarter or higher (no "0"),
a dime (no "15"), or a nickel (no "20"). So 75 is too many
pennies.

If we use fewer pennies, we need even more other coins,
and it doesn't work:
70 pennies: we need 7 more coins, but at most we can use 6 nickels.
65 pennies: we need 12 more coins, but at most we can use 7 nickels.
60 pennies: we need 17 more coins, but at most we can use 8 nickels.
and so on.

So 77 is impossible.

For similar reasons, 81, 85, 86, 89, 90, 93, 94, 95, 97, 98, 99
are also impossible, but all other numbers are possible.

Here is the whole list.
As stated earlier, while many numbers of coins can be achieved
in multiple ways,
I only show one in each case.

(Dollars, Halves, Quarters, Dimes, Nickels)
1: (1,0,0,0,0)
2: (0,2,0,0,0)
3: (0,1,2,0,0)
4: (0,0,4,0,0)
5: (0,1,1,2,1)
6: (0,1,0,5,0)
7: (0,0,3,1,3)
8: (0,0,3,0,5)
9: (0,0,1,7,1)

(Dimes, Nickels, Pennies)
  10: (10, 0, 0) 11: (9, 2, 0) 12: (8, 4, 0)
13: (7, 6, 0) 14: (6, 8, 0) 15: (5, 10, 0) 16: (4, 12, 0)
17: (3, 14, 0) 18: (2, 16, 0) 19: (1, 18, 0) 20: (0, 20, 0)
21: (3, 13, 5) 22: (2, 15, 5) 23: (1, 17, 5) 24: (0, 19, 5)
25: (3, 12, 10) 26: (2, 14, 10) 27: (1, 16, 10) 28: (0, 18, 10)
29: (3, 11, 15) 30: (2, 13, 15) 31: (1, 15, 15) 32: (0, 17, 15)
33: (3, 10, 20) 34: (2, 12, 20) 35: (1, 14, 20) 36: (0, 16, 20)
37: (3, 9, 25) 38: (2, 11, 25) 39: (1, 13, 25) 40: (0, 15, 25)
41: (3, 8, 30) 42: (2, 10, 30) 43: (1, 12, 30) 44: (0, 14, 30)
45: (3, 7, 35) 46: (2, 9, 35) 47: (1, 11, 35) 48: (0, 13, 35)
49: (3, 6, 40) 50: (2, 8, 40) 51: (1, 10, 40) 52: (0, 12, 40)
53: (3, 5, 45) 54: (2, 7, 45) 55: (1, 9, 45) 56: (0, 11, 45)
57: (3, 4, 50) 58: (2, 6, 50) 59: (1, 8, 50) 60: (0, 10, 50)
61: (3, 3, 55) 62: (2, 5, 55) 63: (1, 7, 55) 64: (0, 9, 55)
65: (3, 2, 60) 66: (2, 4, 60) 67: (1, 6, 60) 68: (0, 8, 60)
69: (3, 1, 65) 70: (2, 3, 65) 71: (1, 5, 65) 72: (0, 7, 65)
73: (3, 0, 70) 74: (2, 2, 70) 75: (1, 4, 70) 76: (0, 6, 70)
77: fail 78: (2, 1, 75) 79: (1, 3, 75) 80: (0, 5, 75)
81: fail 82: (2, 0, 80) 83: (1, 2, 80) 84: (0, 4, 80)
85: fail 86: fail 87: (1, 1, 85) 88: (0, 3, 85)
89: fail 90: fail 91: (1, 0, 90) 92: (0, 2, 90)
93: fail 94: fail 95: fail 96: (0, 1, 95)
97: fail 98: fail 99: fail 100: (0, 0, 100)