Paper 2, Section I, 3F

(a) Prove that $\log (n !) \sim n \log n$ as $n \rightarrow \infty$.

(b) State Stirling's approximation for $n$ !.

(c) A school party of $n$ boys and $n$ girls travel on a red bus and a green bus. Each bus can hold $n$ children. The children are distributed at random between the buses.

Let $A_{n}$ be the event that the boys all travel on the red bus and the girls all travel on the green bus. Show that

$\mathbb{P}\left(A_{n}\right) \sim \frac{\sqrt{\pi n}}{4^{n}} \text { as } n \rightarrow \infty$

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