1. Tardigrade
  2. Question
  3. Mathematics
  4. The coefficient of x5 in the expansion of (x2-x -2)5 is

Question Error Report

Thank you for reporting, we will resolve it shortly

Back to Question

Q. The coefficient of $x^{5}$ in the expansion of $(x^{2}-x -2)^{5}$ is

Binomial Theorem

Solution:

$(x -2)^{5} (x +1)^{5}$
$= [^{5}C_{0} x^{5} - {^{5}}C_{1} x^{4} \times 2+ . . . ] [^{5}C_{0} + 5C_{1}x+ . . . ]$
$\Rightarrow $ Coefficient of $x^{5}$
$=\,{}^{5}C_{0} ^{5}C_{5}- \,{}^{5}C_{1}\times2\times\,{}^{5}C_{4}+\,{}^{5}C_{2}\times 2^{2}\times \,{}^{5}C_{3} -\,{}^{5}C_{3}\times2^{3}\times \,{}^{5}C_{2}$
$+\,{}^{5}C_{4}\times2^{4}\times\,{}^{5}C_{1}-\,{}^{5}C_{5}\times2^{5}\times\,{}^{5}C_{0}$
$= 1 -5 \times 5 \times 2 + 10 \times 10 \times 4 -10 \times 10 \times 8 + 5 \times 5 \times 16 -32$
$= -81$