If the coefficients of the r^ th , (r+1)^ th and (r+2)^ th terms in the expansion of (1+x)^n are in the ratio 2 : 15 : 70 , find the coefficient of x¹² in the binomial expansion of (x^r + 1 x^r )^n .

Step-by-step solution

Let the coefficients of the r^ th , (r+1)^ th and (r+2)^ th terms be ^ n C_ r-1 , ^ n C_ r and ^ n C_ r+1 respectively. Given the ratio is 2 : 15 : 70 , we have: ^ n C_ r ^ n C_ r-1 = 15 2 n-r+1 r = 15 2 2n - 2r + 2 = 15r 2n - 17r + 2 = 0 (1) And, ^ n C_ r+1 ^ n C_ r = 70 15 = 14 3 n-r r+1 = 14 3 3n - 3r = 14r + 14 3n - 17r - 14 = 0 (2) Subtracting equation (1) from equation (2), we get: (3n - 17r - 14) - (2n - 17r + 2) = 0 n - 16 = 0 n = 16 Substituting n = 16 in equation (1): 2(16) - 17r + 2 = 0 34 = 17r r = 2 No