# Thread: Find range of positive values of x?

1. ## Find range of positive values of x?

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)]

2. ## Binomial Coefficients

Hello fardeen_gen
Originally Posted by fardeen_gen
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)]
If 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.