Properties of binomial coefficient

Probability Level 2

Let n ∈ N > 0 n \in \mathbb{N}_{> 0} and consider the binomial expansion of ( x + y ) n = ∑ k = 0 k = n α k x k y n − k (x+y)^n = \sum\limits_{k=0}^{k=n} \alpha_k x^k y^{n-k} where α k ∈ R \alpha_k \in \mathbb{R}

Calculate ∑ k α k \sum\limits_k \alpha_k and express your answer as a function of n n .

n n e n e^n 1 3 n 3 − n 2 + 8 3 \frac{1}{3}n^3-n^2+\frac{8}{3} 2 n 2^n cos ⁡ ( 2 π n ) + sin ⁡ ( 2 π n ) \cos(2 \pi n)+\sin(2 \pi n) n 2 − n + 2 n^2-n+2

Patrick Bourg by Patrick Bourg , 28, United Kingdom

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1 solution

Rishabh Jain
Mar 21, 2016

Put x = y = 1 x=y=1 . ⟹ 2 n = ∑ k = 0 n α k \implies \boxed{2^n}=\sum_{k=0}^n \alpha_k

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