How many bit strings of length 12 contain
a) exactly three 1s?
b) at most three 1s?
c) at least three 1s?
d) an equal number of 0s and 1s?

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a. 12-choose-3 Since you're choosing three places for a one to go.

b. \[\binom{12}{0} + \binom{12}{1} + \binom{12}{2} +\binom{12}{3}\]

c. \[2^{12}-\left[\binom{12}{0} + \binom{12}{1} + \binom{12}{2}\right]\]

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