In how many different ways can the sum of 73 cents be made up in coins of denominations of 1, 5, 10,& 25 cents

Answer 1

1. 73 pennies
2. 68 pennies, 1 nickel
3. 63 pennies, 2 nickels
4. 63 pennies, 1 dime
5. 58 pennies, 3 nickels
6. 58 pennies, 1 nickel, 1 dime
7. 53 pennies, 4 nickels
8. 53 pennies, 2 nickels, 1 dime
9. 53 pennies, 2 dimes
10. 48 pennies, 5 nickels
11. 48 pennies, 3 nickels, 1 dime
12. 48 pennies, 1 nickel, 2 dimes
13. 48 pennies, 1 quarter
14. 43 pennies, 6 nickels
15. 43 pennies, 4 nickels, 1 dime
16. 43 pennies, 2 nickels, 1 dime
17. 43 pennies, 1 nickel, 1 quarter
18. 43 pennies, 3 dimes
19. 38 pennies, 7 nickels
20. 38 pennies, 5 nickels, 1 dime
21. 38 pennies, 3 nickels, 2 dimes
22. 38 pennies, 1 nickel, 3 dimes
23. 38 pennies, 2 nickels, 1 quarter
24. 38 pennies, 1 dime, 1 quarter
25. 33 pennies, 8 nickels
26. 33 pennies, 6 nickels, 1 dime
27. 33 pennies, 4 nickels, 2 dimes
28. 33 pennies, 2 nickels, 3 dimes
29. 33 pennies, 4 dimes
30. 33 pennies, 3 nickels, 1 quarter
31. 33 pennies, 1 nickel, 1 dime, 1 quarter
32. 28 pennies, 9 nickels
33. 28 pennies, 7 nickels, 1 dime
34. 28 pennies, 5 nickels, 2 dimes
35. 28 pennies, 3 nickels, 3 dimes
36. 28 pennies, 1 nickel, 4 dimes
37. 28 pennies, 4 nickels, 1 quarter
38. 28 pennies, 2 nickels, 1 dime, 1 quarter
39. 28 pennies, 2 dimes, 1 quarter
40. 23 pennies, 10 nickels
41. 23 pennies, 8 nickels, 1 dime
42. 23 pennies, 6 nickels, 2 dimes
43. 23 pennies, 4 nickels, 3 dimes
44. 23 pennies, 2 nickels, 4 dimes
45. 23 pennies, 5 dimes
46. 23 pennies, 5 nickels, 1 quarter
47. 23 pennies, 3 nickels, 1 dime, 1 quarter
48. 23 pennies, 1 nickel, 2 dimes, 1 quarter
49. 23 pennies, 2 quarters
50. 18 pennies, 11 nickels
51. 18 pennies, 9 nickels, 1 dime
52. 18 pennies, 7 nickels, 2 dimes
53. 18 pennies, 5 nickels, 3 dimes
54. 18 pennies, 3 nickels, 4 dimes
55. 18 pennies, 1 nickel, 5 dimes
56. 18 pennies, 6 nickels, 1 quarter
57. 18 pennies, 4 nickels, 1 dime, 1 quarter
58. 18 pennies, 2 nickels, 2 dimes, 1 quarter
59. 18 pennies, 3 dimes, 1 quarter
60. 18 pennies, 1 nickel, 2 quarter
61. 13 pennies, 12 nickels
62. 13 pennies, 10 nickels, 1 dime
63. 13 pennies, 8 nickels, 2 dimes
64. 13 pennies, 6 nickels, 3 dimes
65. 13 pennies, 4 nickels, 4 dimes
66. 13 pennies, 2 nickels, 5 dimes
67. 13 pennies, 6 dimes
68. 13 pennies, 7 nickels, 1 quarter
69. 13 pennies, 5 nickels, 1 dime, 1 quarter
70. 13 pennies, 3 nickels, 2 dimes, 1 quarter
71. 13 pennies, 1 nickel, 3 dimes, 1 quarter
72. 13 pennies, 2 nickels, 2 quarters
73. 13 pennies, 1 dime, 2 quarters
74. 8 pennies, 13 nickels
75. 8 pennies, 11 nickels, 1 dime
76. 8 pennies, 9 nickels, 2 dimes
77. 8 pennies, 7 nickels, 3 dimes
78. 8 pennies, 5 nickels, 4 dimes
79. 8 pennies, 3 nickels, 5 dimes
80. 8 pennies, 1 nickel, 6 dimes
81. 8 pennies, 8 nickels, 1 quarter
82. 8 pennies, 6 nickels, 1 dime, 1 quarter
83. 8 pennies, 4 nickels, 2 dimes, 1 quarter
84. 8 pennies, 2 nickels, 3 dimes, 1 quarter
85. 8 pennies, 4 dimes, 1 quarter
86. 8 pennies, 3 nickels, 2 quarters
87. 8 pennies, 1 nickel, 1 dime, 2 quarters
88. 3 pennies, 14 nickels
89. 3 pennies, 12 nickels, 1 dime
90. 3 pennies, 10 nickels, 2 dimes
91. 3 pennies, 8 nickels, 3 dimes
92. 3 pennies, 6 nickels, 4 dimes
93. 3 pennies, 4 nickels, 5 dimes
94. 3 pennies, 2 nickels, 6 dimes
95. 3 pennies, 7 dimes
96. 3 pennies, 9 nickels, 1 quarter
97. 3 pennies, 7 nickels, 1 dime, 1 quarter
98. 3 pennies, 5 nickels, 2 dimes, 1 quarter
99. 3 pennies, 3 nickels, 3 dimes, 1 quarter
100. 3 pennies, 1 nickel, 4 dimes, 1 quarter
101. 3 pennies, 4 nickels, 2 quarters
102. 3 pennies, 2 nickels, 1 dime, 2 quarters
103. 3 pennies, 2 dimes, 2 quarters

and a partridge in a pear tree.

Answer 2

Your question is equivalent to this: How many combinations of coins of denomination 25, 10, and 5 add up to 70 cents or less.

Each row here shows a number of quarters, a number of dimes, and a number of solutions with those two as givens:

2 2 1
2 1 3
1 4 2
1 3 4
1 2 6
1 1 8
1 0 10
0 7 1
0 6 3
0 5 5
0 4 7
0 3 9
0 2 11
0 1 13
0 0 15

For example, the row 1 3 4 deals with solutions including 1 quarter and 3 dimes. That's 55 cents, the remaining coins can include 0 nickels, 1 nickel, 2 nickels, or 3 nickels, so there are 4 options. Thus the 4.

Summing the numbers in column 3, we have 4 + 30 + 64 = 98.

Answer 3

73 pennies
1 quarter 4 dimes 8 pennies
1 quarter 4 dimes 1 nickel 3 pennies
1 quarter 3 dimes 2 nickels 8 pennies
1 quarter 3 dimes 18 pennies
1 quarter 3 dimes 1 nickel 13 pennies
1 quarter 3 dimes 2 nickels 8 pennies
1 quarter 3 dimes 3 nickels 3 pennies
1 quarter 2 dimes 28 pennies
1 quarter 2 dimes 5 nickels 3 pennies
1 quarter 2 dimes 4 nickels 8 pennies
1 quarter 2 dimes 3 nickels 13 pennies
1 quarter 2 dimes 2 nickels 18 pennies
1 quarter 2 dimes 1 nickel 23 pennies
1 quarter 2 dimes 28 pennies
1 quarter 1 dime 38 pennies
1 quarter 9 nickels 3 pennies
1 quarter 8 nickels 8 pennies
1 quarter 7 nickels 13 pennies
1 quarter 6 nickels 18 pennies
1 quarter 5 nickels 23 pennies
1 quarter 4 nickels 28 pennies
1 quarter 3 nickels 33 pennies
1 quarter 2 nickels 38 pennies
1 quarter 1 nickel 43 pennies
1 quarter 48 pennies
2 quarters 2 dimes 3 pennies
2 quarters 1 dime 2 nickels 3 pennies
2 quarters 1 dime 1 nickel 8 pennies
2 quarters 1 dime 13 pennies
2 quarters 4 nickels 3 pennies
2 quarters 3 nickels 8 pennies
2 quarters 2 nickels 13 pennies
2 quarters 1 nickel 18 pennies
2 quarters 23 pennies
7 dimes 3 pennies
6 dimes 13 pennies
6 dimes 1 nickel 8 pennies
6 dimes 2 nickels 3 pennies
5 dimes 23 pennies
5 dimes 1 nickel 18 pennies
5 dimes 2 nickels 13 pennies
5 dimes 3 nickels 8 pennies
5 dimes 4 nickels 3 pennies
4 dimes 33 pennies
4 dimes 1 nickel 28 pennies
4 dimes 2 nickels 23 pennies
4 dimes 3 nickels 18 pennies
4 dimes 4 nickels 13 pennies
4 dimes 5 nickels 8 pennies
4 dimes 6 nickels 3 pennies
3 dimes 43 pennies
3 dimes 1 nickel 38 pennies
3 dimes 2 nickels 33 pennies
3 dimes 3 nickels 28 pennies
3 dimes 4 nickels 23 pennies
3 dimes 5 nickels 18 pennies
3 dimes 6 nickels 13 pennies
3 dimes 7 nickels 8 pennies
3 dimes 8 nickels 3 pennies
2 dimes 53 pennies
2 dimes 1 nickel 48 pennies
2 dimes 2 nickels 43 pennies
2 dimes 3 nickels 38 pennies
2 dimes 4 nickels 33 pennies
2 dimes 5 nickels 28 pennies
2 dimes 6 nickels 23 pennies
2 dimes 7 nickels 18 pennies
2 dimes 8 nickels 13 pennies
2 dimes 9 nickels 8 pennies
2 dimes 10 nickels 3 pennies
1 dime 63 pennies
1 dime 1 nickel 58 pennies
1 dime 2 nickels 53 pennies
1 dime 3 nickels 48 pennies
1 dime 4 nickels 43 pennies
1 dime 5 nickels 38 pennies
1 dime 6 nickels 33 pennies
1 dime 7 nickels 28 pennies
1 dime 8 nickels 23 pennies
1 dime 9 nickels 18 pennies
1 dime 10 nickels 13 pennies
1 dime 11 nickels 8 pennies
1 dime 12 nickels 3 pennies
14 nickels 3 pennies
13 nickels 8 pennies
12 nickels 13 pennies
11 nickels 18 pennies
10 nickels 23 pennies
9 nickels 28 pennies
8 nickels 33 pennies
7 nickels 38 pennies
6 nickels 43 pennies
5 nickels 48 pennies
4 nickels 53 pennies
3 nickels 58 pennies
2 nickels 63 pennies
1 nickel 68 pennies

98 ways check ya later ♥

Answer 4

92 combinations:

73 pennies
1 nickel 68 pennies
1 dime 63 pennies
2 nickels 63 pennies
1 dime 1 nickel 58 pennies
3 nickels 58 pennies
2 dimes 53 pennies
1 dime 2 nickels 53 pennies
4 nickels 53 pennies
1 quarter 48 pennies
2 dimes 1 nickel 48 pennies
1 dime 3 nickels 48 pennies
5 nickels 48 pennies
1 quarter 1 nickel 43 pennies
3 dimes 43 pennies
2 dimes 2 nickels 43 pennies
1 dime 4 nickels 43 pennies
6 nickels 43 pennies
1 quarter 1 dime 38 pennies
1 quarter 2 nickels 38 pennies
3 dimes 1 nickel 38 pennies
2 dimes 3 nickels 38 pennies
1 dime 5 nickels 38 pennies
7 nickels 38 pennies
1 quarter 1 dime 1 nickel 33 pennies
1 quarter 3 nickels 33 pennies
4 dimes 33 pennies
3 dimes 2 nickels 33 pennies
2 dimes 4 nickels 33 pennies
1 dime 6 nickels 33 pennies
8 nickels 33 pennies
1 quarter 2 dimes 28 pennies
1 quarter 1 dime 2 nickels 28 pennies
1 quarter 3 nickels 28 pennies
4 dimes 1 nickel 28 pennies
3 dimes 3 nickels 28 pennies
2 dimes 5 nickels 28 pennies
1 dime 7 nickels 28 pennies
9 nickels 28 pennies
2 quarters 23 pennies
1 quarter 2 dimes 1 nickel 23 pennies
1 quarter 1 dime 3 nickels 23 pennies
1 quarter 5 nickels 23 pennies
5 dimes 23 pennies
4 dimes 2 nickels 23 pennies
3 dimes 4 nickels 23 pennies
2 dimes 6 nickels 23 pennies
1 dime 8 nickels 23 pennies
10 nickels 23 pennies
2 quarters 1 nickel 18 pennies
1 quarter 3 dimes 18 pennies
1 quarter 2 dimes 2 nickels 18 pennies
1 quarter 1 dime 4 nickels 18 pennies
1 quarter 6 nickels 18 pennies
5 dimes 1 nickel 18 pennies
4 dimes 3 nickels 18 pennies
3 dimes 5 nickels 18 pennies
2 dimes 7 nickels 18 pennies
1 dime 9 nickels 18 pennies
11 nickels 18 pennies
2 quarters 1 dime 13 pennies
2 quarters 2 nickels 13 pennies
1 quarter 3 dimes 1 nickel 13 pennies
1 quarter 2 dimes 3 nickels 13 pennies
1 quarter 1 dime 5 nickels 13 pennies
1 quarter 7 nickels 13 pennies
6 dimes 13 pennies
5 dimes 2 nickels 13 pennies
4 dimes 4 nickels 13 pennies
3 dimes 6 nickels 13 pennies
2 dimes 8 nickels 13 pennies
10 nickels 13 pennies
2 quarters 1 dime 1 nickel 8 pennies
2 quarters 3 nickels 8 pennies
1 quarter 4 dimes 8 pennies
1 quarter 3 dimes 2 nickels 8 pennies
1 quarter 2 dimes 4 nickels 8 pennies
1 quarter1 dime 6 nickels 8 pennies
1 quarter 8 nickels 8 pennies
6 dimes 1 nickel 8 pennies
5 dimes 3 nickels 8 pennies
4 dimes 5 nickels 8 pennies
3 dimes 7 nickels 8 pennies
2 dimes 9 nickels 8 pennies
1 dime 11 nickels 8 pennies
13 nickels 8 pennies
2 quarters 2 dimes 3 pennies
2 quarters 1 dime 2 nickels 3 pennies
2 quarters 4 nickels 3 pennies
1 quarter 4 dimes 1 nickel 3 pennies
1 quarter 3 dimes 3 nickels 3 pennies
1 quarter 2 dimes 5 nickels 3 pennies
1 quarter 1 dime 7 nickels 3 pennies
1 quarter 9 nickels 3 pennies
7 dimes 3 pennies
6 dimes 2 nickels 3 pennies
5 dimes 4 nickels 3 pennies
4 dimes 6 nickels 3 pennies
3 dimes 8 nickels 3 pennies
2 dimes 10 nickels 3 pennies
1 dime 12 nickels 3 pennies
14 nickels 3 pennies

Well clearly I missed something

Answer 5

Since 73 is not a multiple of 25, 10, or 5, clearly there must be SOME pennies in any solution.

There is 1 way to make 73¢ out of pennies.
There are 14 ways to make 73¢ out of pennies and nickels.
There are 7 ways to make 73¢ out of pennies and dimes.
There are 2 ways to make 73¢ out of pennies and quarters.
There are 6+5+4+3+2+1=21 ways to make 73¢ out of pennies, nickels, and dimes. (for each way to make 73¢ out of pennies and dimes, replace a dime with 2 nickels until only one dime is left)
There are 4+9=13 ways to make 73¢ out of pennies, nickels and quarters (if there are 2 quarters you can replace pennies with nickels 4 times, if there is 1 quarter, 9 times.)
There are 2+4=6 ways to make 73¢ out of pennies, dimes and quarters (if there are 2 quarters you can replace pennies with dimes 2 times. if 1 quarter, 4 times)
There are 1+7+5+3=16 ways to make 73¢ out of pennies, nickels, dimes, and quarters. (If there are 2q and 2d, you can replace 1d with 2n. If there are 1q and 4d, you can replace up to 3d and up to 5p with 1,2,3,4,5,6, or 7n. If there are 1q and 3d you can replace up to 2d and up to 5p without overlapping the 1q + 4d case, with 1,2,3,4, or 5n. If there are 1q and 2d you can replace 1d and/or 5p with 1,2, or 3n.)

n = 1 + 14 + 7 + 2 + 21 + 13 + 6 + 16

n = 80 <<<<<<<<<<<

blahb is clearly right and I am clearly wrong. I will figure out what I missed and add it in later, but I don't have time now.

-----------------
EDIT:
For my PND case there are actually 42 not 21 possibilities. For each case where you replace n dimes with 2n nickels, there should also be a case where you replace n dimes each with (2n-1) nickels and 5 pennies.

For my PNDQ case there are actually 2 + 7 + 5 + 3 + 1 = 18 possibilities, not 1 + 7 + 5 + 3. Starting from 3p0n2d2q you can replace 1d with 2n or with 5p1n. Starting from 8p0n4d1q there are 7 possibilities (as I noted). Likewise with 18p0n3d1q (5) and 28p0n2d1q (3). But I failed to note that starting from 38p0n1d1q there is on possibility of converting 5p into 1n.

So my revised final result is:

n = 1 + 14 + 7 + 2 + 42 + 13 + 6 + 18
n = 103 <<<<<<<

which matches blahb's CLEARLY CORRECT result.

Actually there is one small mistake (clearly a typo) in blahb's result. #16 should have 2 dimes, not 1.

-----------------------
ASIDE TO THOSE WHO ARE GETTING 98/92...
You are missing something somewhere :(

Grant: you are missing the 2q 0d case (for which there are 5 possibilities)

jimbob and dabears: Your enumerations are not sorted in any logical order. That makes it hard to see what you are missing.

Answer 6

Heh! I can just see people all over the world sitting at their computer desks with the contents of their change jars spread out and trying this one out!

Answer 7

I would say 12 ways

Answer 8

lots, I suggest makeing a chart,

4 rows, and then make all teh combinations, I think there would be about a hundred.

Answer 9

do you know the anwser of that question? because i have entirely no clue.

if you do know, could you pleezew tell me the answer. i have a question under arts & humanities > philosophy >"why are people bad?" you dont need to answer my question, just tell me the answer to yours.

Answer 10

I dont even have time to figure that out.