Question

Find the coefficient of $x^{40}$ in the expansion of $(1+2x+x^2{)}^{27}$.

• A. ${54}_{C_{40}}$
• B. ${44}_{C_{10}}$
• C. ${54}_{C_{27}}$
• D. None of these

Answer 1

The correct option is A ${54}_{C_{40}}$

Here,

$(1+2x+x^2{)}^{27}=[(1+x{)}^2{]}^{27}=(1+x{)}^{54}$

Suppose $x^{40}$ occurs in $(r+1)th$ term in the expansion of $(1+x{)}^{54}$.

Now, $T_{r+1}={54}_{C_r}{x}^r$

For this term to contain $x^{40}$, we must have $r=40$.

So, coefficient of $x^{40}={54}_{C_{40}}$