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7.Binomial Theorem
hard
If for positive integers $r > 1,n > 2$ the coefficient of the ${(3r)^{th}}$ and ${(r + 2)^{th}}$ powers of $x$ in the expansion of ${(1 + x)^{2n}}$ are equal, then
A
$n = 2r$
B
$n = 3r$
C
$n = 2r + 1$
D
None of these
(AIEEE-2002) (IIT-1983)
Solution
(c) In the expansion of $(1+x)^{2n}$, the general term
$ = {\,^{2n}}{C_k},0 \le k \le 2n$
As given for $r > 1,n > 2{,^{2n}}{C_{3r}} = {\,^{2n}}{C_{r + 2}}$
==> Either $3r = r + 2$
or $3r = 2n – (r + 2)$,
==> $r = 1$ or $n = 2r + 1 \Rightarrow n = 2r + 1$, .
Standard 11
Mathematics
