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Today's calculation of Integral 568

Source: Kyoto University entrance exam/Science A, Problem 6

February 26, 2010
calculusintegrationprobabilitylimitlogarithmscalculus computations

Problem Statement

Throw n n balls in to 2n 2n boxes.　Suppose each ball comes into each box with equal probability of entering in any boxes. Let pn p_n be the probability such that any box has ball less than or equal to one. Find the limit lim⁡n→∞ln⁡pnn \lim_{n\to\infty} \frac{\ln p_n}{n}
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