Question

Sa se determine n care verifica egalitatea : $(C2n)^{0}+(C2n)^{1} +(C2n)^{2}+...+(C2n)^{2n} =1024$

Answer 1

$\text{Suma coeficientilor binomiali:}\\C_{n}^0+C_{n}^1+C_{n}^2+...+C_{n}^{n} = 2^{n}\\ \\ \Rightarrow C_{2n}^0+C_{2n}^1+C_{2n}^2+...+C_{2n}^{2n} = 2^{2n} \\ \\ 2^{2n} = 1024 \Rightarrow 2^{2n} = 2^{10} \Rightarrow 2n = 10 \Rightarrow \boxed{n =5}$