In the expansion of x 3 - 1 x 2 n , n ∈ N , if, the sum of the coefficients of x 5 and x 10 is 0 , then n is:

We know that the general term of the binomial expansion of a + b n is given by T r ' + 1 = C r ' n a n - r ' b r ' . Therefore, general term of x 3 - 1 x 2 n is given by T r + 1 = C r n x 3 n - r - 1 x 2 r ⇒ T r + 1 = C r n x 3 n - 3 r - 1 r x - 2 r ⇒ T r + 1 = C r n x 3 n - 5 r - 1 r For coefficient of x 5 and x 10 , substitute 3 n - 5 r = 5 and 10 , respectively, Hence, coefficient of x 5 is C 3 n - 5 5 n - 1 3 n - 5 5 And coefficient of x 10 is C 3 n - 10 5 n - 1 3 n - 10 5 Now, Coefficient of x 5 + Coefficient of x 10 = 0 C 3 n - 5 5 n - 1 3 n - 5 5 + C 3 n - 10 5 n - 1 3 n - 10 5 = 0 ⇒ C 3 n - 5 5 n - 1 3 n - 5 5 = C 3 n - 10 5 n - 1 3 n - 5 5 ⇒ C 3 n - 5 5 n = C 3 n - 10 5 n ⇒ 3 n - 5 5 + 3 n - 10 5 = n ⇒ 6 n - 15 5 = n ⇒ 6 n - 15 = 5 n ∴ n = 15

Find the middle term(s) in the expansion of x y - y x 7 .

Given, x y - y x 7 There should be two middle terms. T 4 , T 5 → Middle term T 4 = c 3 7 x y 4 - y x 3 T 4 = - 35 x y T 5 = c 4 7 x y 3 y x 4 = 35 x y