If the coefficient of a 7 b 8 in the expansion of ( a + 2 b + 4 a b ) 10 is K · 2 16 , then K is equal to

Step-by-step solution

Given the expansion     ( a + 2 b + 4 a b ) 10 T r + 1   = a 10 · b 10 1 b + 2 a + 4 10 T r + 1 = a 10 · b 10     C 10 r 1 + r 2   1 b r 1 2 a r 2 4 10 - r 1 - r 2 T r + 1 = a 10 · b 10 10 ! 1 b r 1 2 a r 2 4 10 - r 1 - r 2 r 1 ! r 2 ! 10 - r 1 - r 2 ! Putting, r 1 = 2 ,   r 2 = 3 Coefficient of a 7 b 8 is 10 ! · 2 3 · 4 10 - 2 - 3 2 ! . 3 ! 5 ! = 2 13 · 10 ! 2 ! . 3 ! 5 ! = 2 16 · 315 K . 2 16 = 315 . . 2 16 Hence, K = 315