Question

For $n\in N,(1+x{)}^n-nx-1$ is divisible by

• A. $2$
• B. $x$
• C. $x^2$
• D. $x^3$

Answer 1

Answer: (B), (C)

The correct options are
B $x$
C $x^2$

We have,
$(1+x{)}^n-nx-1$
${=}^n{C}_2\left(x{)}^2{+}^n{C}_3\right(x{)}^3+...{+}^n{C}_{n-1}\left(x{)}^{n-1}{+}^n{C}_n\right(x{)}^n$
$=x^2\left({}^n{C}_2{+}^n{C}_3\right(x)+...{+}^n{C}_{n-1}(x{)}^{n-3}{+}^n{C}_n\left(x{)}^{n-2}\right)$
Hence, options B and C are the correct answers.