Question

Find the coeff of $x^5$ in the expansion of the product $(1+2x{)}^5(1-x{)}^7$

• A. $139$
• B. $155$
• C. $171$
• D. $-109$

Answer 1

The correct option is C $171$

For $(1+2x{)}^5$, General term, $T_{r+1}{=}^5{C}_r{\left(2x\right)}^r$

For $(1-x{)}^7$, General term, $T_{r+1}{=}^7{C}_r{(-x)}^r$

Now, we can get 5 in 6 ways: $0+51+42+33+24+15+0$

So, we have to make use of these combinatons.

Hence coefficient of $x^5=$

$1\times {(-1)}^7{C_5{+}^5{C}_1.2.}^7{C_4-}^5{C_2.2^2.}^7{C_3+}^5{C_3.2^3.}^7{C_2{-}^5{C_4.2^4.}^7{C}_1+}^5{C_5.2^5.}^7{C}_0$

$=171$