lf some three consecutive coefficients in the binomial expansion of (x + 1)n in powers of x are in the ratio 2 : 15 : 70, then the average of these three coefficients is

Given binomial is (x+ 1)n, whose general term, is Tr+1=nCrxrAccording to the question, we have nCr−1:nCr:nCr+1=2:15:70now, nCr−1 nCr=215⇒n!(r−1)!(n−r+1)!n!r!(n−r)!=215⇒rn−r+1=215⇒ 15r=2n−2r+2⇒ 2n−17r+2=0----i⇒  similarly,   nCr nCr+1=1570⇒n!r!(n−r)!n!(r+1)!(n−r−1)!=314⇒ r+1n−r=314⇒14r+14=3n−3r⇒ 3n−17r−14=0----iiOn solving Eqs. (i) and (ii), we getn−16=0⇒n=16 and r=2Now, the average = 16C1+16C2+16C33                           =16+120+5603=6963=232