In the expansion of (1 + x)^18 if the term with greatest Binomial coefficient is also the numerically greatest term, then range of positive values of x is:
A) [(10/11),(11/10)]
B) [(9/10), (10/9)]
C) [(9/11), (11/9)]
D) [(8/9), (9/8)]
In the expansion of (1 + x)^18 if the term with greatest Binomial coefficient is also the numerically greatest term, then range of positive values of x is:
A) [(10/11),(11/10)]
B) [(9/10), (10/9)]
C) [(9/11), (11/9)]
D) [(8/9), (9/8)]
Hello fardeen_genIf n is even, the greatest Binomial coefficient of $\displaystyle \binom n r $ is the one in the middle; i.e. where $\displaystyle r = \frac{n}{2}$ (If n is odd, it's either of the two equal terms in the middle.) So here the greatest coefficient is $\displaystyle \binom {18}{9}$.
So, if this term also has the greatest value, then
$\displaystyle \binom{18}{8}x^8<\binom{18}{9}x^9$, and
$\displaystyle \binom{18}{10}x^{10}<\binom{18}{9}x^9$
$\displaystyle \Rightarrow x>\frac{\binom{18}{8}}{\binom{18}{9}}$, and
$\displaystyle \Rightarrow x<\frac{\binom{18}{9}}{\binom{18}{10}}$
$\displaystyle \Rightarrow \frac{9}{10}<x<\frac{10}{9}$
So the answer is B.
Grandad