Paper 4, Section II, 8E

Numbers and Sets | Part IA, 2016

Define the binomial coefficient (nm)\left(\begin{array}{c}n \\ m\end{array}\right). Prove directly from your definition that

(1+z)n=m=0n(nm)zm(1+z)^{n}=\sum_{m=0}^{n}\left(\begin{array}{c} n \\ m \end{array}\right) z^{m}

for any complex number zz.

(a) Using this formula, or otherwise, show that

k=03n(3)k(6n2k)=26n\sum_{k=0}^{3 n}(-3)^{k}\left(\begin{array}{l} 6 n \\ 2 k \end{array}\right)=2^{6 n}

(b) By differentiating, or otherwise, evaluate m=0nm(nm)\sum_{m=0}^{n} m\left(\begin{array}{c}n \\ m\end{array}\right).

Let Sr(n)=m=0n(1)mmr(nm)S_{r}(n)=\sum_{m=0}^{n}(-1)^{m} m^{r}\left(\begin{array}{c}n \\ m\end{array}\right), where rr is a non-negative integer. Show that Sr(n)=0S_{r}(n)=0 for r<nr<n. Evaluate Sn(n)S_{n}(n).

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