Phương pháp giải:

+) Xuất phát từ khai triển nhị thức \({\left( {a + b} \right)^n} = C_n^0{a^n} + C_n^1{a^{n - 1}}b + C_n^2{a^{n - 2}}{b^2} + ... + C_n^{n - 1}a{b^{n - 1}} + C_n^n{b^n}\)

+) Thay \(a,b,n\) bằng các giá trị thích hợp.

Lời giải chi tiết:

Ta có: \({\left( {a + b} \right)^n} = C_n^0{a^n} + C_n^1{a^{n - 1}}b + C_n^2{a^{n - 2}}{b^2} + ... + C_n^{n - 1}a{b^{n - 1}} + C_n^n{b^n}\)

Thay \(a = 1,b = 1\) ta có:

\({2^n} = C_n^0 + C_n^1 + C_n^2 + C_n^3 + ... + C_n^{n - 2} + C_n^{n - 1} + C_n^n\)

Đáp án A đúng.

Ta có: \({\left( {a + b} \right)^{2n}} = C_{2n}^0{a^{2n}} + C_{2n}^1{a^{2n - 1}}b + C_{2n}^2{a^{2n - 2}}{b^2} + ... + C_{2n}^{2n - 1}a{b^{2n - 1}} + C_{2n}^{2n}{b^{2n}}\)

Thay \(a = 1,b = 1\) ta có:

\(\begin{array}{l}
{2^{2n}} = C_{2n}^0 + C_{2n}^1 + C_{2n}^2 + C_{2n}^3 + ... + C_{2n}^{2n - 2} + C_{2n}^{2n - 1} + C_{2n}^{2n}\\
\Leftrightarrow C_{2n}^0 + C_{2n}^1 + C_{2n}^2 + C_{2n}^3 + ... + C_{2n}^{2n - 2} + C_{2n}^{2n - 1} + C_{2n}^{2n} = {\left( {{2^2}} \right)^n}\\
\Leftrightarrow C_{2n}^0 + C_{2n}^1 + C_{2n}^2 + C_{2n}^3 + ... + C_{2n}^{2n - 2} + C_{2n}^{2n - 1} + C_{2n}^{2n} = {4^n}
\end{array}\)

Đáp án B đúng.

Ta có: \({\left( {a + b} \right)^{3n}} = C_{3n}^0{a^{3n}} + C_{3n}^1{a^{3n - 1}}b + C_{3n}^2{a^{3n - 2}}{b^2} + ... + C_{3n}^{3n - 1}a{b^{3n - 1}} + C_{3n}^{3n}{b^{3n}}\)

Thay \(a = 1,b = 1\) ta có:

\(\begin{array}{l}
{2^{3n}} = C_{3n}^0 + C_{3n}^1 + C_{3n}^2 + C_{3n}^3 + ... + C_{3n}^{3n - 2} + C_{3n}^{3n - 1} + C_{3n}^{3n}\\
\Leftrightarrow C_{3n}^0 + C_{3n}^1 + C_{3n}^2 + C_{3n}^3 + ... + C_{3n}^{3n - 2} + C_{3n}^{3n - 1} + C_{3n}^{3n} = {\left( {{2^3}} \right)^n}\\
\Leftrightarrow C_{3n}^0 + C_{3n}^1 + C_{3n}^2 + C_{3n}^3 + ... + C_{3n}^{3n - 2} + C_{3n}^{3n - 1} + C_{3n}^{3n} = {8^n}
\end{array}\)

Đáp án C sai.

Ta có: \({\left( {a + b} \right)^{4n}} = C_{4n}^0{a^{4n}} + C_{4n}^1{a^{4n - 1}}b + C_{4n}^2{a^{4n - 2}}{b^2} + ... + C_{4n}^{4n - 1}a{b^{4n - 1}} + C_{4n}^{4n}{b^{4n}}\)

Thay \(a = 1,b = 1\) ta có:

\(\begin{array}{l}
{2^{4n}} = C_{4n}^0 + C_{4n}^1 + C_{4n}^2 + C_{4n}^3 + ... + C_{4n}^{4n - 2} + C_{4n}^{4n - 1} + C_{4n}^{4n}\\
\Leftrightarrow C_{4n}^0 + C_{4n}^1 + C_{4n}^2 + C_{4n}^3 + ... + C_{4n}^{4n - 2} + C_{4n}^{4n - 1} + C_{4n}^{4n} = {\left( {{2^4}} \right)^n}\\
\Leftrightarrow C_{4n}^0 + C_{4n}^1 + C_{4n}^2 + C_{4n}^3 + ... + C_{4n}^{4n - 2} + C_{4n}^{4n - 1} + C_{4n}^{4n} = {16^n}
\end{array}\)

Đáp án D đúng.

Chọn C.