Question

For natural number $m,n$, if $(1-y{)}^m(1-y{)}^n=1+a_{1}y+a_2{y}^2+........$ 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)

$(1-y{)}^m(1+y{)}^n=1+a_{1}y+a_2{y}^2+.......$
$\Rightarrow (1{-}^m{C}_{1}y{+}^m{C}_2{y}^2-...........)(1{+}^n{C}_{1}y{+}^n{C}_2{y}^2+.......)=1+a_{1}y+a_2{y}^2+...........$
$\Rightarrow a_1{=}^n{C}_1{-}^m{C}_1,a_2{=}^m{C}_2{-}^m{C}_1^n{C}_1{+}^n{C}_2$
$\Rightarrow n-m=10,m^2+n^2-m-n-2mn=20$
$\Rightarrow n-m=10,(m-n{)}^2-(m+n)=20$
$\Rightarrow n-m=10,n+m=80$
$\Rightarrow m=35,n=45$