Subsets of a set

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Consider a set of $10$ elements.

(a) How many subsets does it have in all (including the empty set and the set itself)?

(b) How many of these subsets have exactly $3$ elements?

Solution

(a) For each element we independently decide whether to include it ($1$) or not ($0$): these are $10$ binary choices, i.e. $N_{\text{tot}} = 2^{10} = \boxed{1024}.$

(b) The $3$-element subsets are the combinations of $10$ elements taken $3$ at a time: $N_{3} = \binom{10}{3} = \frac{10\cdot 9\cdot 8}{6} = \boxed{120}.$