Number of topping combinations from ten toppings

Question

An ice cream shop offers $10$ different toppings, and a dessert may include any selection of them as long as at least one topping is used. How many distinct topping combinations can be made?

Short answer

$2^{10}-1=1023$ . Each topping is independently either on or off, giving $2^{10}$ subsets; discard the single empty subset because at least one topping is required. Equivalently, $\sum_{k=1}^{10}\binom{10}{k}=1023$ .

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Lessons behind this question

Counting and Combinatorics

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