$\text{ For natural numbers }\mathbf{m},\mathbf{n}\text{ if }(\mathbf{l}-\mathbf{y}{)}^m(\mathbf{l}+\mathbf{y}{)}^{\mathbf{n}}=\mathbf{1}+{\mathbf{a}}_1\mathbf{y}+{\mathbf{a}}_2{\mathbf{y}}^2+\dots ,\text{ and }{\mathbf{a}}_{\mathbf{l}}={\mathbf{a}}_{\mathbf{2}}=\mathbf{10}\text{ , }$then (m, n) is

$\begin{array}{l}(1-y{)}^m(1+y{)}^n=\left(1-{}^m{C}_{1}y+{}^{m}C_2{y}^2-\dots \right)\left(1+{}^{n}C_{1}y+{}^{n}C_2{y}^2+\dots .\right)=1+a_1\text{y }+a_2{y}^2+\dots ..\\ \\ \Rightarrow a_1={}^{n}C_1-{}^{m}C_1, a_2={}^{m}C_2-{}^{m}C_1{}^{n}C_1+{}^{n}C_2\Rightarrow n-m=10, m^2+n^2-m-n-2mn=20\\ \Rightarrow n-m=10, (m-n{)}^2-(m+n)=20\\ \Rightarrow n-m=10, (m+n)=80\Rightarrow m=35, n=45\\ \therefore \left(m,n\right)=\left(35,45\right)\end{array}$