Given the integers r > 1, n > 2, and coefficient of

Question:

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

(a) n = 2r

(b) n = 3r

(c) n = 2r + 1

(d) none of these

Solution:

Given r > 1 and n > 2

and coefficient of T3r = coefficient of Tr+2 is expansion of (1 + x)2n

i.e. ${ }^{2 n} C_{3 r-1}={ }^{2 n} C_{r+1}$

i. e. $3 r-1=r+1$ (using property of combination)

and $2 n-3 r+1=r+1$

i. e. $2 r=2$ and $2 n=4 r$

i. e. $r=1$ and $n=2 r$

Hence, the correct answer is option A.

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