There are 100 people in a room, all of them wearing different hats. Everybody puts their hat in a large basket.
Later, after thoroughly shuffling the hats, each person takes out one hat and wears it.
Let the probability of a randomly picked person wearing their own hat be .
What is , where denotes the greatest integer less than or equal to .
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The probability of the first person picking his/her own hat = 1 0 0 1
Probability of second person picking their own hat = (probability of 1st person not picking second person's hat)*(probability of second person picking their own hat out of the remaining 99) = 1 0 0 9 9 9 9 1 = 1 0 0 1
Going on, it can be shown that probability of n-th person picking their own hat = 1 0 0 × 9 9 × ⋯ ( 1 0 0 − n + 1 ) 9 9 × 9 8 × ⋯ ( 1 0 0 − n ) = 1 0 0 1
Hence, everyone has a probability of {0.01=1%) or x = 1
So, [ 1 0 0 0 x ] = 1 0 0 0