Question

The coeffecients of the middle term in the binomial expansion in powers of $x$ of ${(1+\alpha x)}^4$ and ${(1+\alpha x)}^6$ is the same if $\alpha$ equals

• A. $-\frac{5}{3}$
• B. $\frac{10}{3}$
• C. $\frac{3}{10}$
• D. $\frac{3}{5}$

Answer 1

The correct option is C $\frac{3}{10}$

The middle terms of the above given binomial expressions will be
${}^4{C}_2(\alpha {)}^2{x}^2$ and ${}^6{C}_3(\alpha {)}^3{x}^3$
Since the coefficients are equal,
${}^4{C}_2(\alpha {)}^2={}^6{C}_3(\alpha {)}^3$
$6(\alpha {)}^2=20(\alpha {)}^3$
$\frac{3}{10}=\alpha$