For the natural numbers m, n, if ( 1 - y ) ^ m ( 1 + y ) ^ n = 1 +...

We have,

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

$= \left(C0 m - C1 m y + C2 m {y}^{2} + . . . + Cm m {\left(- 1\right)}^{m}{y}^{m}\right)\left(C0 n + C1 n y + C2 n {y}^{2} + . . . + Cn n {y}^{n}\right)$

Coefficient of $y\left({a}_{1}\right) = 1 \cdot C1 n + C1 m ( - 1 )$

$= n - m = 10 \ldots \left(1\right)$

Coefficient of ${y}^{2}\left({a}_{2}\right)$

$= 1 C2 n - C1 m C1 n + {1}^{m}{C}_{2} = 10$

$⇒ \dfrac{n ( n - 1 )}{2} - m \cdot n + \dfrac{m ( m - 1 )}{2} = 10$

$⇒ {m}^{2} + {n}^{2} - 2mn - ( n + m ) = 20$

$⇒ ( n - m{)}^{2} - ( n + m ) = 20$

$⇒ n + m = 80 \ldots$\left(2\right)$</p><p>By equation \left(1\right)$ and \left(2\right)$</p><p>$m = 35, n = 45$</p><p>Hence,$m + n = 80