 Home

Codex Wiki

List of universities

University of Cambridge

Course of the University of Cambridge

Mathematics course of the University of Cambridge

Past exam of the mathematics course of the University of Cambridge

2019

Paper 2

Every allocation is determined by the set of $n$ children on the red bus, so there are $\binom{2n}{n}$ equally likely allocations. Exactly one places every boy on red and every girl on green. Hence
$$
\mathbb P(A_n)=\binom{2n}{n}^{-1}=\frac{(n!)^2}{(2n)!}.
$$
Applying <Stirling formula> to the numerator and denominator gives the <central binomial coefficient> asymptotic
$$
\binom{2n}{n}\sim\frac{4^n}{\sqrt{\pi n}},
$$
and therefore
$$
\boxed{\mathbb P(A_n)\sim\frac{\sqrt{\pi n}}{4^n}}.
$$

Solution

Past exam of the mathematics course of the University of Cambridge / 2019 / ia / Paper 2 / 3F / c / Solution

Past exam of the mathematics course of the University of Cambridge / 2019 / ia / Paper 2 / 3F / c / Solution

 Ancestors (11)

 Discussion (0)

 Articles by others on the same topic (0)

Past exam of the mathematics course of the University of Cambridge / 2019 / ia / Paper 2 / 3F / c / Solution

 Ancestors (11)

 Discussion (0)  Subscribe (1)

 Articles by others on the same topic (0)

 Discussion (0)