Difference between 办法， 方法， and 方式？
I haven't looked up in the dictionary, so my answer may not be accurate. Just for reference.
方法 means the way that should be followed when doing something. Example: 使用方法 (the way to use something, like washing machine or ointment)
办法 seems to be more colloquial, and mostly refers to the way to solve a problem or achieve a goal. Sometimes it is equivalent to 方法. Example: 我没办法吃这么多。(I can't make it to eat this much. )
方式 refers to the "style" when doing something. However, it is not eqivalent to 样式. The latter refers to what a thing looks like. Example: 传统方式未必是落后的。(Traditional way(style) is not definitely out of date. ) And: 传统的样式是很好看的。(Traditional style is very beautiful. )
Hope this will help you. ^o^