Given positive integers r1, n2 and the coefficient of (3 r) th and (r+2) th terms in the binomial expansion of (1+x)2 n are equal, then

Question

Given positive integers r>1, n>2 and the coefficient of (3 r) th and (r+2) th terms in the binomial expansion of (1+x)2 n are equal, then (A) n=2r (B) n=2r +1 (C) n=3r (D) None of these

Tardigrade Admin · Accepted Answer

3 r th term in the expansion of (1+x)2 n = 2 n C3 r-1 x3 r-1 and (r+2) th term in the expansion of (1+x)2 n = 2 n Cr+1 xr+1 Given that the binomial coefficients of (3 r) th and (r+2) th terms are equal. ∴ 2 n C3 r-1= 2 n Cr+1 ⇒ 3 r-1=r+1 or 2 n=(3 r-1)+(r+1) ⇒ 2 r=2 or 2 n=4 r ⇒ r=1 or n=2 r But r>1 ∴ n=2 r

Q. Given positive integers r>1,n>2 and the coefficient of (3r) th and (r+2) th terms in the binomial expansion of (1+x)2n are equal, then

n=2r

n=2r +1

n=3r

None of these

Q. Given positive integers $r > 1, n > 2$ and the coefficient of $(3 r)$ th and $(r + 2)$ th terms in the binomial expansion of $(1 + x)^{2 n}$ are equal, then