# SOLUTION: how many ways to make change for one dollar using nickels, dimes and quarters

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 Click here to see ALL problems on Word Problems With Coins Question 152564: how many ways to make change for one dollar using nickels, dimes and quartersFound 3 solutions by Edwin McCravy, pmr2teach, richard1234:Answer by Edwin McCravy(8909)   (Show Source): You can put this solution on YOUR website!``` We can use either 0, 1, 2, 3, or 4 quarters. If we use 0 quarters, we can use from 0 up to 10 dimes, and the rest, if any, in nickels. That accounts for 11 ways. If we use 1 quarter, we can use from 0 up to 7 dimes, and the rest in nickels. That accounts for 8 ways. If we use 2 quarters, we can use from 0 up to 5 dimes, and the rest, if any, in nickels. That accounts for 6 ways. If we use 3 quarters, we can use from 0 up to 2 dimes, and the rest in nickels. That accounts for 3 ways. If we use 4 quarters, that's the whole dollar, so that accounts for 1 way. So the total number of ways = 11+8+6+3+1 = 29 You weren't asked to list them, but here is the list of all 29 ways: 1. 0 quarters, 0 dimes, and 20 nickels. 2. 0 quarters, 1 dime, and 18 nickels. 3. 0 quarters, 2 dimes, and 16 nickels. 4. 0 quarters, 3 dimes, and 14 nickels. 5. 0 quarters, 4 dimes, and 12 nickels. 6. 0 quarters, 5 dimes, and 10 nickels. 7. 0 quarters, 6 dimes, and 8 nickels. 8. 0 quarters, 7 dimes, and 6 nickels. 9. 0 quarters, 8 dimes, and 4 nickels. 10. 0 quarters, 9 dimes, and 2 nickels. 11. 0 quarters, 10 dimes, and 0 nickels. 12. 1 quarter, 0 dimes, and 15 nickels. 13. 1 quarter, 1 dime, and 13 nickels. 14. 1 quarter, 2 dimes, and 11 nickels. 15. 1 quarter, 3 dimes, and 9 nickels. 16. 1 quarter, 4 dimes, and 7 nickels. 17. 1 quarter, 5 dimes, and 5 nickels. 18. 1 quarter, 6 dimes, and 3 nickels. 19. 1 quarter, 7 dimes, and 1 nickel. 20. 2 quarters, 0 dimes, and 10 nickels. 21. 2 quarters, 1 dime, and 8 nickels. 22. 2 quarters, 2 dimes, and 6 nickels. 23. 2 quarters, 3 dimes, and 4 nickels. 24. 2 quarters, 4 dimes, and 2 nickels. 25. 2 quarters, 5 dimes, and 0 nickels. 26. 3 quarters, 0 dimes, and 5 nickels. 27. 3 quarters, 1 dime, and 3 nickels. 28. 3 quarters, 2 dimes, and 1 nickel. 29. 4 quarters, 0 dimes, and 0 nickels. Edwin``` Answer by pmr2teach(1)   (Show Source): You can put this solution on YOUR website!Actually there are over 293 ways to use coins to make a dollar!!!! They are as follows: 1 dollar coin 2 half dollars 1 HD 2Q 1 HD 1Q 2D IN 1 HD 1Q 2D 5P 1 HD 1Q 1D 3N 1 HD 1Q 1D 2N 5P 1 HD 1Q 1D 1N 10P 1 HD 1Q 1D 15P 1 HD 1Q 5N 1 HD 1Q 4N 5P 1 HD 1Q 3N 10P 1 HD 1Q 2N 15P 1 HD 1Q 1N 20P 1 HD 1Q 25P 1 HD 5D 1 HD 4D 2N 1 HD 4D 1N 5P 1 HD 4D 10P 1 HD 3D 4N 1 HD 3D 3N 5P 1 HD 3D 2N 10P 1 HD 3D 1N 15P 1 HD 3D 20P 1 HD 2D 6N 1 HD 2D 5N 5P 1 HD 2D 4N 10P 1 HD 2D 3N 15P 1 HD 2D 2N 20P 1 HD 2D 1N 25P 1 HD 2D 30P 1 HD 1D 8N 1 HD 1D 7N 5P 1 HD 1D 6N 10P 1 HD 1D 5N 15P 1 HD 1D 4N 20P 1 HD 1D 3N 25 P 1 HD 1D 2N 30 P 1 HD 1D 1N 35P 1 HD 1D 40P 1 HD 10N 1 HD 9N 5P 1 HD 8N 10P 1 HD 7N 15P 1 HD 6N 20P 1 HD 5N 25P 1 HD 4N 30P 1 HD 3N 35P 1 HD 2N 40P 1 HD 1N 45P 1 HD 50P 4Q 3Q 2D 1N 3Q 2D 5P 3Q 1D 3N 3Q 1D 2N 5P 3Q 1D 1N 10P 3Q 1D 15P 3Q 5N 3Q 4N 5P 3Q 3N 10P 3Q 2N 15P 3Q 1N 20P 3Q 25P 2Q 5D 2Q 4D 2N 2Q 4D 1N 5P 2Q 4D 10P 2Q 3D 4N 2Q 3D 3N 5P 2Q 3D 2N 10P 2Q 3D 1N 15P 2Q 3D 20P 2Q 2D 6N 2Q 2D 5N 5P 2Q 2D 4N 10P 2Q 2D 3N 15P 2Q 2D 2N 20P 2Q 2D 1N 25P 2Q 2D 30P 2Q 1D 8N 2Q 1D 7N 5P 2Q 1D 6N 10P 2Q 1D 5N 15P 2Q 1D 4N 20P 2Q 1D 3N 25P 2Q 1D 2N 30P 2Q 1D 1N 35P 2Q 1D 40P 2Q 50P 2Q 10N 2Q 9N 5P 2Q 8N 10P 2Q 7N 15P 2Q 6N 20P 2Q 5N 25P 2Q 4N 30P 2Q 3N 35P 2Q 2N 40P 2Q 1N 45P 1Q 7D 1N 1Q 7D 5P 1Q 6D 3N 1Q 6D 2N 5P 1Q 6D 1N 10P 1Q 6D 15P 1Q 5D 5N 1Q 5D 4N 5P 1Q 5D 3N 10P 1Q 5D 2N 15P 1Q 5D 1N 20P 1Q 5D 25P 1Q 4D 7N 1Q 4D 6N 5P 1Q 4D 5N 15P 1Q 4D 4N 20P 1Q 4D 3N 25P 1Q 4D 2N 30P 1Q 4D 1N 35P 1Q 4D 40P 1Q 3D 9N 1Q 3D 8N 5P 1Q 3D 7N 10P 1Q 3D 6N 15P 1Q 3D 5N 20P 1Q 3D 4N 25P 1Q 3D 3N 30P 1Q 3D 2N 35P 1Q 3D 1N 40P 1Q 3D 45P 1Q 2D 11N 1Q 2D 10N 5P 1Q 2D 9N 10P 1Q 2D 8N 15P 1Q 2D 7N 20P 1Q 2D 6N 25P 1Q 2D 5N 30P 1Q 2D 4N 35P 1Q 2D 3N 40P 1Q 2D 2N 45P 1Q 2D 1N 50P 1Q 2D 55P 1Q 1D 13N 1Q 1D 12N 5P 1Q 1D 11N 10P 1Q 1D 10N 15P 1Q 1D 9N 20P 1Q 1D 8N 25P 1Q 1D 7N 30P 1Q 1D 6N 35P 1Q 1D 5N 40P 1Q 1D 4N 45P 1Q 1D 3N 50P 1Q 1D 2N 55P 1Q 1D 1N 60P 1Q 1D 65P 1Q 15N 1Q 14N 5P 1Q 13N 10P 1Q 12N 15P 1Q 11N 20P 1Q 10N 25P 1Q 9N 30P 1Q 8N 35P 1Q 7N 40P 1Q 6N 45P 1Q 5N 50P 1Q 4N 55P 1Q 3N 60P 1Q 2N 65P 1Q 1N 70P 1Q 75P 10D 9D 2N 9D 1N 5P 9D 10P 8D 4N 8D 3N 5P 8D 2N 10P 8D 1N 15P 8D 20P 7D 6N 7D 5N 5P 7D 4N 10P 7D 3N 15P 7D 2N 20P 7D 1N 25P 7D 30P 6D 8N 6D 7N 5P 6D 6N 10P 6D 5N 15P 6D 4N 20P 6D 3N 25P 6D 2N 30P 6D 1N 35P 6D 40P 5D 10N 5D 9N 5P 5D 8N 10P 5D 7N 15P 5D 6N 20P 5D 5N 25P 5D 4N 30P 5D 3N 35P 5D 2N 40P 5D 1N 45P 5D 50P 4D 12N 4D 11N 5P 4D 10N 10P 4D 9N 15P 4D 8N 20P 4D 7N 25P 4D 6N 30P 4D 5N 35P 4D 4N 40P 4D 3N 45P 4D 2N 50P 4D 1N 55P 4D 60P 3D 14N 3D 13N 5P 3D 12N 10P 3D 11N 15P 3D 10N 20P 3D 9N 25P 3D 8N 30P 3D 7N 35P 3D 6N 40P 3D 5N 45P 3D 4N 50P 3D 3N 55P 3D 2N 60P 3D 1N 65P 3D 70P 2D 16N 2D 15N 5P 2D 14N 10P 2D 13N 15P 2D 12N 20P 2D 11N 25P 2D 10N 30P 2D 9N 35P 2D 8N 40P 2D 7N 45P 2D 6N 50P 2D 5N 55P 2D 4N 60P 2D 3N 65P 2D 2N 70P 2D 1N 75P 2D 80P 1D 18N 1D 17N 5P 1D 16N 10P 1D 15N 15P 1D 14N 20P 1D 13N 25P 1D 12N 30P 1D 11N 35P 1D 10N 40P 1D 9N 45P 1D 8N 50P 1D 7N 55P 1D 6N 60P 1D 5N 65P 1D 4N 70P 1D 3N 75P 1D 2N 80P 1D 1N 85P 1D 90P 20N 19N 5P 18N 10P 17N 15P 16N 20P 15N 25P 14N 30P 13N 35P 12N 40P 11N 45P 10N 50P 9N 55P 8N 60P 7N 65P 6N 70P 5N 75P 4N 80P 3N 85P 2N 90P 1N 95P 100P Read more: http://wiki.answers.com/Q/How_many_ways_to_make_change_for_a_dollar#ixzz1IfkbKEMU Answer by richard1234(5390)   (Show Source): You can put this solution on YOUR website!You can compute the number of ways to make change for \$1 using a bijection. We let n, d, q be the number of nickels, dimes, and quarters, and , which implies . Hence, this is equivalent to finding the number of ways to make 20 "cents" using one-, two-, and five-cent pieces. Suppose we want to find the number of ways to make 20 cents *without* using a five cent piece. Case 0: We wish to make 0 cents --> 1 way (just use no coins) Case 1: We wish to make 1 cent --> 1 way 2*0 + 1*1 Case 2: We wish to make 2 cents --> 2 ways 2*1 + 1*0; 2*0 + 1*2 Case 3: We wish to make 3 cents --> 2 ways 2*1 + 1*1; 2*0 + 1*3 Case 4: We wish to make 4 cents --> 3 ways 2*2 + 1*0; 2*1 + 1*2, 2*0 + 1*4 Case 5: We wish to make 5 cents --> 3 ways 2*2 + 1*1, 2*1 + 1*3, 2*0 + 1*5 We can prove using either induction or modular arithmetic that the number of ways to make n cents in this way is , where denotes the floor value of x. We can evaluate this at n = 20 to get , or 11. Similarly, we can count the number of ways to obtain 20 cents using one five-cent piece. However, we can subtract off the five-cent piece and say that this is analogous to computing the number of ways to obtain 15 cents. Hence, this is equal to . For 10, 5, and 0 cents, we have , and . Therefore the total number of ways is 11 + 8 + 6 + 3 + 1 = 29 ways. Note: the other tutor counted 293 ways, however this included pennies and half dollars, which was not stated in the question. To count the number of ways using pennies, nickels, dimes, and quarters, you can also use a bijection, but instead of counting the number of ways to obtain 20 cents, you evaluate at 20, 19, 18, ..., 0 since this will uniquely determine the number of pennies. We do the same with 15, 10, 5, and 0 to obtain sums: Then, if you want half dollars involved, it is equal to where is the number of ways to make a dollar using i half dollars, equivalently, the number of ways to make 50 cents and 0 cents without half dollars (here you should get 293!).