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293 ways.Using pennies, nickels, dimes, quarters, half-dollars and dollar coins, there are 293 ways to make change for one dollar in US currency.

Method 1:

Using pennies, nickels, dimes, quarters, and half-dollars there are 292 ways to make change for one dollar. In general, to make change for h half-dollars using these coins there are (6 + 55*h + 119*h2 + 95*h3 + 25*h4)/6 ways. For example, putting h=1 gives 50 ways of making change for fifty cents, or h=2 gives 292 ways of making change for dollar etc.

[[ The 292 ways computed above, includes fifty-cent pieces but not the dollar coin so, we must add one more in order to agree with the second method shown below. ]]

Method 2:

Take the Taylor expansion of 1 over (1-x)*(1-x5)*(1-x10)*(1-x25)*(1-x50)*(1-x100), and jot down the coefficient of x100. This is the generating function for the ways to partition 100 into subsets of size 1, 5, 10, 25, 50, or 100, so there are 293 ways to make change for one us dollar.

[[ if we use no dollar coins or half dollars (only pennies, nickels, dimes and quarters) the method 2 gives 242 ways to make a dollar. Those 242 ways begin with line 52. ]]

Indisputable Tally of Combinations
  1. 1 dollar coin
  2. 2 half dollars
  3. 1 HD 2Q
  4. 1 HD 1Q 2D IN
  5. 1 HD 1Q 2D 5P
  6. 1 HD 1Q 1D 3N
  7. 1 HD 1Q 1D 2N 5P
  8. 1 HD 1Q 1D 1N 10P
  9. 1 HD 1Q 1D 15P
  10. 1 HD 1Q 5N
  11. 1 HD 1Q 4N 5P
  12. 1 HD 1Q 3N 10P
  13. 1 HD 1Q 2N 15P
  14. 1 HD 1Q 1N 20P
  15. 1 HD 1Q 25P
  16. 1 HD 5D
  17. 1 HD 4D 2N
  18. 1 HD 4D 1N 5P
  19. 1 HD 4D 10P
  20. 1 HD 3D 4N
  21. 1 HD 3D 3N 5P
  22. 1 HD 3D 2N 10P
  23. 1 HD 3D 1N 15P
  24. 1 HD 3D 20P
  25. 1 HD 2D 6N
  26. 1 HD 2D 5N 5P
  27. 1 HD 2D 4N 10P
  28. 1 HD 2D 3N 15P
  29. 1 HD 2D 2N 20P
  30. 1 HD 2D 1N 25P
  31. 1 HD 2D 30P
  32. 1 HD 1D 8N
  33. 1 HD 1D 7N 5P
  34. 1 HD 1D 6N 10P
  35. 1 HD 1D 5N 15P
  36. 1 HD 1D 4N 20P
  37. 1 HD 1D 3N 25 P
  38. 1 HD 1D 2N 30 P
  39. 1 HD 1D 1N 35P
  40. 1 HD 1D 40P
  41. 1 HD 10N
  42. 1 HD 9N 5P
  43. 1 HD 8N 10P
  44. 1 HD 7N 15P
  45. 1 HD 6N 20P
  46. 1 HD 5N 25P
  47. 1 HD 4N 30P
  48. 1 HD 3N 35P
  49. 1 HD 2N 40P
  50. 1 HD 1N 45P
  51. 1 HD 50P
  52. 4Q
  53. 3Q 2D 1N
  54. 3Q 2D 5P
  55. 3Q 1D 3N
  56. 3Q 1D 2N 5P
  57. 3Q 1D 1N 10P
  58. 3Q 1D 15P
  59. 3Q 5N
  60. 3Q 4N 5P
  61. 3Q 3N 10P
  62. 3Q 2N 15P
  63. 3Q 1N 20P
  64. 3Q 25P
  65. 2Q 5D
  66. 2Q 4D 2N
  67. 2Q 4D 1N 5P
  68. 2Q 4D 10P
  69. 2Q 3D 4N
  70. 2Q 3D 3N 5P
  71. 2Q 3D 2N 10P
  72. 2Q 3D 1N 15P
  73. 2Q 3D 20P
  74. 2Q 2D 6N
  75. 2Q 2D 5N 5P
  76. 2Q 2D 4N 10P
  77. 2Q 2D 3N 15P
  78. 2Q 2D 2N 20P
  79. 2Q 2D 1N 25P
  80. 2Q 2D 30P
  81. 2Q 1D 8N
  82. 2Q 1D 7N 5P
  83. 2Q 1D 6N 10P
  84. 2Q 1D 5N 15P
  85. 2Q 1D 4N 20P
  86. 2Q 1D 3N 25P
  87. 2Q 1D 2N 30P
  88. 2Q 1D 1N 35P
  89. 2Q 1D 40P
  90. 2Q 50P
  91. 2Q 10N
  92. 2Q 9N 5P
  93. 2Q 8N 10P
  94. 2Q 7N 15P
  95. 2Q 6N 20P
  96. 2Q 5N 25P
  97. 2Q 4N 30P
  98. 2Q 3N 35P
  99. 2Q 2N 40P
  100. 2Q 1N 45P
  101. 1Q 7D 1N
  102. 1Q 7D 5P
  103. 1Q 6D 3N
  104. 1Q 6D 2N 5P
  105. 1Q 6D 1N 10P
  106. 1Q 6D 15P
  107. 1Q 5D 5N
  108. 1Q 5D 4N 5P
  109. 1Q 5D 3N 10P
  110. 1Q 5D 2N 15P
  111. 1Q 5D 1N 20P
  112. 1Q 5D 25P
  113. 1Q 4D 7N
  114. 1Q 4D 6N 5P
  115. 1Q 4D 5N 15P
  116. 1Q 4D 4N 20P
  117. 1Q 4D 3N 25P
  118. 1Q 4D 2N 30P
  119. 1Q 4D 1N 35P
  120. 1Q 4D 40P
  121. 1Q 3D 9N
  122. 1Q 3D 8N 5P
  123. 1Q 3D 7N 10P
  124. 1Q 3D 6N 15P
  125. 1Q 3D 5N 20P
  126. 1Q 3D 4N 25P
  127. 1Q 3D 3N 30P
  128. 1Q 3D 2N 35P
  129. 1Q 3D 1N 40P
  130. 1Q 3D 45P
  131. 1Q 2D 11N
  132. 1Q 2D 10N 5P
  133. 1Q 2D 9N 10P
  134. 1Q 2D 8N 15P
  135. 1Q 2D 7N 20P
  136. 1Q 2D 6N 25P
  137. 1Q 2D 5N 30P
  138. 1Q 2D 4N 35P
  139. 1Q 2D 3N 40P
  140. 1Q 2D 2N 45P
  141. 1Q 2D 1N 50P
  142. 1Q 2D 55P
  143. 1Q 1D 13N
  144. 1Q 1D 12N 5P
  145. 1Q 1D 11N 10P
  146. 1Q 1D 10N 15P
  147. 1Q 1D 9N 20P
  148. 1Q 1D 8N 25P
  149. 1Q 1D 7N 30P
  150. 1Q 1D 6N 35P
  151. 1Q 1D 5N 40P
  152. 1Q 1D 4N 45P
  153. 1Q 1D 3N 50P
  154. 1Q 1D 2N 55P
  155. 1Q 1D 1N 60P
  156. 1Q 1D 65P
  157. 1Q 15N
  158. 1Q 14N 5P
  159. 1Q 13N 10P
  160. 1Q 12N 15P
  161. 1Q 11N 20P
  162. 1Q 10N 25P
  163. 1Q 9N 30P
  164. 1Q 8N 35P
  165. 1Q 7N 40P
  166. 1Q 6N 45P
  167. 1Q 5N 50P
  168. 1Q 4N 55P
  169. 1Q 3N 60P
  170. 1Q 2N 65P
  171. 1Q 1N 70P
  172. 1Q 75P
  173. 10D
  174. 9D 2N
  175. 9D 1N 5P
  176. 9D 10P
  177. 8D 4N
  178. 8D 3N 5P
  179. 8D 2N 10P
  180. 8D 1N 15P
  181. 8D 20P
  182. 7D 6N
  183. 7D 5N 5P
  184. 7D 4N 10P
  185. 7D 3N 15P
  186. 7D 2N 20P
  187. 7D 1N 25P
  188. 7D 30P
  189. 6D 8N
  190. 6D 7N 5P
  191. 6D 6N 10P
  192. 6D 5N 15P
  193. 6D 4N 20P
  194. 6D 3N 25P
  195. 6D 2N 30P
  196. 6D 1N 35P
  197. 6D 40P
  198. 5D 10N
  199. 5D 9N 5P
  200. 5D 8N 10P
  201. 5D 7N 15P
  202. 5D 6N 20P
  203. 5D 5N 25P
  204. 5D 4N 30P
  205. 5D 3N 35P
  206. 5D 2N 40P
  207. 5D 1N 45P
  208. 5D 50P
  209. 4D 12N
  210. 4D 11N 5P
  211. 4D 10N 10P
  212. 4D 9N 15P
  213. 4D 8N 20P
  214. 4D 7N 25P
  215. 4D 6N 30P
  216. 4D 5N 35P
  217. 4D 4N 40P
  218. 4D 3N 45P
  219. 4D 2N 50P
  220. 4D 1N 55P
  221. 4D 60P
  222. 3D 14N
  223. 3D 13N 5P
  224. 3D 12N 10P
  225. 3D 11N 15P
  226. 3D 10N 20P
  227. 3D 9N 25P
  228. 3D 8N 30P
  229. 3D 7N 35P
  230. 3D 6N 40P
  231. 3D 5N 45P
  232. 3D 4N 50P
  233. 3D 3N 55P
  234. 3D 2N 60P
  235. 3D 1N 65P
  236. 3D 70P
  237. 2D 16N
  238. 2D 15N 5P
  239. 2D 14N 10P
  240. 2D 13N 15P
  241. 2D 12N 20P
  242. 2D 11N 25P
  243. 2D 10N 30P
  244. 2D 9N 35P
  245. 2D 8N 40P
  246. 2D 7N 45P
  247. 2D 6N 50P
  248. 2D 5N 55P
  249. 2D 4N 60P
  250. 2D 3N 65P
  251. 2D 2N 70P
  252. 2D 1N 75P
  253. 2D 80P
  254. 1D 18N
  255. 1D 17N 5P
  256. 1D 16N 10P
  257. 1D 15N 15P
  258. 1D 14N 20P
  259. 1D 13N 25P
  260. 1D 12N 30P
  261. 1D 11N 35P
  262. 1D 10N 40P
  263. 1D 9N 45P
  264. 1D 8N 50P
  265. 1D 7N 55P
  266. 1D 6N 60P
  267. 1D 5N 65P
  268. 1D 4N 70P
  269. 1D 3N 75P
  270. 1D 2N 80P
  271. 1D 1N 85P
  272. 1D 90P
  273. 20N
  274. 19N 5P
  275. 18N 10P
  276. 17N 15P
  277. 16N 20P
  278. 15N 25P
  279. 14N 30P
  280. 13N 35P
  281. 12N 40P
  282. 11N 45P
  283. 10N 50P
  284. 9N 55P
  285. 8N 60P
  286. 7N 65P
  287. 6N 70P
  288. 5N 75P
  289. 4N 80P
  290. 3N 85P
  291. 2N 90P
  292. 1N 95P
  293. 100P

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