Menu Combinations

Algebra 2 · Axiom Academy

A cafe has a few appetizers, a few mains, a few desserts. How many different full meals is that? The answer hides a rule that runs all through counting and probability. At Cafe Combinations you build a meal by choosing one appetizer, then one main, then one dessert. It is tempting to just add up the options, but the number of different meals grows much faster than that. Watch the count fan out. Each appetizer can be followed by any main, and each of those by any dessert, so every choice branches into all the next ones. Watch the tree grow: 3 appetizers, then mains, then desserts, fanning out to complete meals. Multiplication is the branching: each of the 3 appetizers opens into 4 mains (that is 12), and each of those into 2 desserts (that is 24). Change the menu, watch the total Drag the sliders to change how many appetizers, mains, and desserts the cafe offers. The tree and the total update instantly, and the total is always the product . Add one more dessert and the whole total jumps, because that new dessert pairs with every appetizer-and-main combination that already existed. Pick one appetizer, one main, and one dessert. Your choice lights up a single path from Start down to one leaf, one meal out of all 24 . Every leaf on the tree is a different meal you could have built. That is the whole idea: the total counts the paths, and any specific meal is exactly one of them. The fundamental counting principle

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