Question

For natural numbers $m,n$ if $(1-y{)}^m(1+y{)}^n=1+a_{1}y+a_2{y}^2+\dots$ and $a_1=a_2=10$ then $(m,n)$ is:

• A. $(45,35)$
• B. $(35,45)$
• C. $(20,45)$
• D. $(35,20)$

Answer 1

Answer: (B)

$(35,45)$

$(1-y{)}^m(1+y{)}^n$

$=(1-my+{}^m{C}_2{y}^2...(-1{)}^m{}^m{C}_m{y}^m\left)\right(1+ny+{}^n{C}_2{y}^2...+{}^n{C}_n{y}^m)$

Therefore coefficient of $y$ is $n-m$ ...(i)

Coefficient of $y^2$ is ${}^n{C}_2-mn+{}^m{C}_2$ ..(ii)

Therefore, $n-m=10$ and ${}^n{C}_2-mn+{}^m{C}_2=10$

$\Rightarrow \frac{n(n-1)}{2}+\frac{m(m-1)}{2}-mn=10$

$\Rightarrow \frac{1}{2}(n^2-2mn+m^2-(m+n\left)\right)=10$

$\Rightarrow (m-n{)}^2-(m+n)=20$

$\Rightarrow 100-(m+n)=20$

$\Rightarrow m+n=80$ And

$\Rightarrow n-m=10$

Hence $2n=90$

$n=45$ and therefore $m=35$

Hence, $(m,n)=(35,45)$