A gambit is a kind of sacrifice, while a gamut is a range. Your offer to do more chores might be a gambit to get a higher allowance — but if your parents simply pile on more tasks without offering more cash, your feelings might run the gamut from sad to angrily disappointed.

A gambit is an opening remark or offer that you hope will help you later. It came into English as a chess term, as in the Queen's gambit, an opening move in which you sacrifice a Queen's pawn in order to set up a win later on. This word has since moved beyond the chess board, as in these examples:

Many saw the bill as a negotiating gambit to win concessions from the EU in wider negotiations on a trade deal. (Reuters)

If the gambit works, savings squeezed from the pharmaceutical industry would eventually end up covering the cost of the copay cards. (Washington Times)

Gamut, on the other hand, is a range from one extreme to the other. It originally meant the lowest musical note on the scale but now it refers to all the notes from low to high (or high to low) — or to any similar range. These days, it's most often used in the phrase "run the gamut," as in the following examples:

The results run the gamut from clever to outlandish. (Salon)

Residents run the gamut from young families to retirees. (Washington Post)

To remember the difference, a gambit is like a tricky bet, and gamut has a "u" for run the gamut.