Question

The sum of the third from the beginning and the third from the end of the binomial coefficients in the expansion of ${(\sqrt[4]{43}+\sqrt[3]{34})}^n$ is equal to $9900.$ The number of rational terms contained in the expansion is:

• A. $8$
• B. $9$
• C. $10$
• D. $11$

Answer 1

The correct option is B $9$

We have ${}^n{C}_2{+}^n{C}_{n-2}=9900$
Also,
${}^n{C}_2={}^n{C}_{n-2}$
$\Rightarrow {}^n{C}_2=4950$
$\Rightarrow \frac{n(n-1)}{2}=4950$
$\Rightarrow n^2-n-9900=0$
On solving we get, $n=100$.
Now, we wish to find the rational terms.
Hence, $\frac{r}{4}$ and $\frac{100-r}{3}$ must be integers.
For the second condition,
$r=1,4,7,10,13,16,...,99$
$r$ must also be divisible by $4.$
Hence, $r$ can take the values : $4,16,28,40,52,64,76,88,100$.
Hence, there are $9$ terms which are rational.