9.20 Find the largest binomial coefficient in the expansion of $\left(n+\frac{1}{n}\right)^{n}$, if the product of the fourth term from the beginning and the fourth term from the end is 14400.
9.20 The fourth term from the beginning has the form $T_{4}=C_{n}^{3} n^{n-3} \frac{1}{n^{3}}$, and the fourth term from the end has the form $T_{n-2}=C_{n}^{n-3} n^{3} \frac{1}{n^{n-3}}$.

Therefore, $T_{4} T_{n-2}=\left(C_{n}^{3}\right)^{2}=14400$, from which $C_{n}^{3}=120$. Further, we have

$$
n(n-1)(n-2)=720 ; n(n-1)(n-2)=10 \cdot 9 \cdot 8 ; n=10
$$

Thus, the largest binomial coefficient, which is equally distant from the ends of the expansion, is $C_{10}^{5}=252$.

Answer: 252.