3F (Probability)

Since the natural logarithm is increasing, comparison with an integral gives

\int_{1}^{n} lo g x d x \leq \sum_{k = 1}^{n} lo g k \leq n lo g n .

The lower bound is

n lo g n - n + 1

. Dividing all three expressions by

n lo g n

and applying the squeeze theorem gives

lo g (n!) \sim n lo g n .

b

n! \sim 2 πn (\frac{n}{e})^{n} .

Every allocation is determined by the set of

n

children on the red bus, so there are

(n 2 n)

equally likely allocations. Exactly one places every boy on red and every girl on green. Hence

P (A_{n}) = (n 2 n)^{- 1} = \frac{( n ! ) ^{2}}{( 2 n )!} .

(n 2 n) \sim \frac{4 ^{n}}{πn},

and therefore

P (A_{n}) \sim \frac{πn}{4 ^{n}} .

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