Question

Find the coefficient of ${\text{x}}^5$ in ${\left(1+2\text{x}\right)}^6+{\left(1-\text{x}\right)}^7$ .

Answer 1

General term is given by ${}^n{C}_r{x}^r$

In First term the term containing $x^5$ is given by

${}^6{C}_5\left(2x{)}^5\frac{6!}{5!}\right(32)x^{5}6\times 32x^{5}192x^5\cdots (1)$

In Second term the term containing $x^5$ is given by

${}^7{C}_5(-x{)}^5\frac{7!}{5!}(-x^5)7\times 6\times -x^5-42x^5\cdots (2)$

On adding both $192-42=150$

So the coefficient of $x^5$ in given expression is $150$