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Year 4 of the mathematics course of the University of Oxford

Graph Theory and Probabilistic Combinatorics course of the University of Oxford

Past exam of the graph Theory and Probabilistic Combinatorics course of the University of Oxford

2008

For $k\ge1$, monotonicity of the logarithm gives
$$
\log(k!)=\sum_{j=1}^k\log j\ge\int_1^k\log t\,dt=k\log k-k+1\ge k\log k-k.
$$
Thus $k!\ge(k/e)^k$. Bounding the numerator of the <binomial coefficient> by $n^k$ yields
$$
\boxed{\binom nk\le\frac{n^k}{k!}\le(en/k)^k}.
$$

Solution

Past exam of the graph Theory and Probabilistic Combinatorics course of the University of Oxford / 2008 / B / 5 / a / i / Solution

Past exam of the graph Theory and Probabilistic Combinatorics course of the University of Oxford / 2008 / B / 5 / a / i / Solution

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