Probability of picking 4 right people out of 12 There are 12 people with 12 unique zodiac signs. You start guessing each person's zodiac sign. What is the probability of you guessing at least 4 of them right?
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You can't repeat your guess (you can only say "you are libra" once).
 A: Remember, there are $n!$ ways to arrange $n$ distinct items in a row.
Remember further that you can decide a "correct order" to arrange them in (for example, people from youngest to oldest, or star signs from January to December).
Finally, remember that there are $!n$ ways to arrange those $n$ items in a row such that no item is in its "correct" spot where $!n$ is the Recontres number counting the number of derangements of $n$ objects.

To continue your problem, note that to have guessed exactly $k$ correct star-signs (given that each person has a distinct star-sign and you guess each star-sign exactly once, etc...) requires that you have guessed correctly $k$ star-signs for some subset of $k$ of the people while having guessed incorrectly for the remaining $n-k$ people.

 The probability of guessing exactly $k$ correct is then $\dfrac{\binom{n}{k}\cdot !(n-k)}{n!}$

Complete the problem, noting that to have guessed at least $4$ right involves guessing exactly $4$ right, or guessing exactly $5$ right, or ...  or alternatively not having guessed exactly $0$ right or exactly $1$ right etc...
