Let A and B be the coefficients of the 8^ th and 16^ th terms respectively in the binomial expansion of (1+x)^n . If 34A = 5B , then the coefficient of x^4 in the expansion of (x + 1 x^2 )^n is :

Step-by-step solution

A = ^ n C₇ and B = ^ n C₁₅ 34 ^ n C₇ = 5 ^ n C₁₅ 34 n! 7!(n-7)! = 5 n! 15!(n-15)! (n-7)! (n-15)! = 34 5 15! 7! (n-7)(n-8) (n-14) = 34 5 15 14 13 12 11 10 9 8 (n-7)(n-8) (n-14) = 102 14 13 12 11 10 9 8 (n-7)(n-8) (n-14) = 18 17 16 15 14 13 12 11 Equating the largest factors on both sides: n-7 = 18 n = 25 For the expansion of (x + 1 x^2 )²⁵ , the general term is: T_ r+1 = ²⁵C_ r x^ 25-r (x⁻²)^r = ²⁵C_ r x^ 25-3r To find the coefficient of x^4 , we set the exponent of x to 4 : 25 - 3r = 4 3r = 21 r = 7 Thus, the requi