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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?
 2 years ago
 2 years ago
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?
 2 years ago
 2 years ago

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KingGeorgeBest ResponseYou've already chosen the best response.2
a. 12choose3 Since you're choosing three places for a one to go.
 2 years ago

KingGeorgeBest ResponseYou've already chosen the best response.2
b. \[\binom{12}{0} + \binom{12}{1} + \binom{12}{2} +\binom{12}{3}\]
 2 years ago

KingGeorgeBest ResponseYou've already chosen the best response.2
c. \[2^{12}\left[\binom{12}{0} + \binom{12}{1} + \binom{12}{2}\right]\]
 2 years ago

KingGeorgeBest ResponseYou've already chosen the best response.2
If you want an explanation for any one of these, please ask.
 2 years ago

inkyvoydBest ResponseYou've already chosen the best response.0
Wrong, all of those. I'm pretty sure that that format means combination, while you're looking for permutations.
 2 years ago

windsylphBest ResponseYou've already chosen the best response.0
i think that the problem seeks combinations actually, i just don't know how to come up with these numbers.
 2 years ago

windsylphBest ResponseYou've already chosen the best response.0
or wait.. okay im confused. haha. permutations because order matters in bit strings?
 2 years ago

inkyvoydBest ResponseYou've already chosen the best response.0
No, Permutations because order matters. 111000000000 is different from 011100000000 is it not?
 2 years ago

windsylphBest ResponseYou've already chosen the best response.0
haha okay thanks for clearing that up.
 2 years ago

inkyvoydBest ResponseYou've already chosen the best response.0
If you ever get confused about whether to use permutations or combinations, just remember the order matters rule for permutations.
 2 years ago

KingGeorgeBest ResponseYou've already chosen the best response.2
Sorry, had to step away for a little. You definitely want combinations here. Suppose you have this string: 111000000000. If you were using permutations, you would have \(3! \cdot 9!\) ways to make this string. However, there is only one string that looks like this, not \(3!\cdot9!\) strings. In this problem, order does not matter because we're considering the order of the ones with relation to the other ones. Since \(1=1\), order does not matter.
 2 years ago
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