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:

Expanding the terms:$(1-my+\frac{m(m-1)}{2}{y}^2\dots )(1+ny+\frac{n(n-1)}{2}{y}^2\dots )=1+a_{1}y+a_2{y}^2\dots$ Coefficient of $y$: $a_1=n-m=10$ Coefficient of $y^2$: $a_2=\frac{n(n-1)}{2}+\frac{m(m-1)}{2}-mn=10$$n^2-n+m^2-m-2mn=20$$(n-m{)}^2-(n+m)=20$ Since $n-m=10$:$10^2-(n+m)=20⟹100-(n+m)=20⟹n+m=80$ We have:$n-m=10$$n+m=80$ Adding the equations: $2n=90⟹n=45$.Subtracting: $2m=70⟹m=35$.So, $(m,n)=(35,45)$.