For the natural numbers m , n , if ( 1 - y ) m ( 1 + y ) n = 1 + a 1 y + a 2 y 2 + … . + a m + n y m + n and a 1 = a 2 = 10 , then the value of m + n , is equal to:

Step-by-step solution

We have, ( 1 - y ) m ( 1 + y ) n = C 0 m - C 1 m y + C 2 m y 2 + . . . + C m m - 1 m y m C 0 n + C 1 n y + C 2 n y 2 + . . . + C n n y n Coefficient of y a 1 = 1 · C 1 n + C 1 m ( - 1 ) = n - m = 10   … 1 Coefficient of y 2 a 2 = 1 C 2 n - C 1 m C 1 n + 1 m C 2 = 10 ⇒ n ( n - 1 ) 2 - m · n + m ( m - 1 ) 2 = 10 ⇒ m 2 + n 2 - 2 m n - ( n + m ) = 20 ⇒ ( n - m ) 2 - ( n + m ) = 20 ⇒ n + m = 80   … 2 By equation 1 and 2 m = 35 ,   n = 45 Hence, m + n = 80