Tính tổng \(S = {3^{2015}}.C_{2015}^0 - {3^{2014}}.C_{2015}^2 + {3^{2013}}.C_{2015}^2 - ... + 3C_{2015}^{2014} - C_{2015}^{2015}\).

Đáp án đúng là: A

Theo nhị thức Newton ta có:

\({(3 + x)^{2015}} = C_{2015}^0{.3^{2015}} + C_{2015}^1{.3^{2014}}.x + C_{2015}^2{.3^{2013}}.{x^2} + \ldots . + C_{2015}^{2014}.3.{x^{2014}} + C_{2015}^{2015}.{x^{2015}}\)

Thay x = –1 ta được:

\({(3 - 1)^{2015}} = C_{2015}^0{.3^{2015}} - C_{2015}^1{.3^{2014}} + C_{2015}^2{.3^{2013}} - \ldots . + C_{2015}^{2014}.3 - C_{2015}^{2015}\)

Suy ra, \(S = {2^{2015}}\)

Vậy ta chọn đáp án A.