If \(A_{1},A_{2},A_{3}.....A_{1006}\) be **independent events** such that \(P(A_{i})=\frac{1}{2i}(i=1,2,3....1006)\) and probability that **none of the events** occurs be \(\frac{\alpha!}{2^{\alpha}(\beta!)^{2}}\), then

\(A)\)- \(\beta\) is of form \(4k+2\) where \(k\) is a positive integer.

\(B)\)- \(\alpha=2\beta\)

\(C)\)- \(\beta\) is a composite number

\(D)\)- \(\alpha\) is of form \(4k\) where where \(k\) is a positive integer.

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