**Solution:**

To find the number of solutions of the equation $\sin(x) \sin(3x) \sin(5x) = 0$ in the interval $\left[\frac{\pi}{2}, \frac{3\pi}{2}\right]$, we need to analyze the behavior of each factor individually and identify the values of $x$ for which the equation is satisfied.

**Analyzing the First Factor: $\sin(x)$**

The function $\sin(x)$ is equal to zero at $x = \frac{\pi}{2}$ and $x = \frac{3\pi}{2}$. However, since the interval of interest is $\left[\frac{\pi}{2}, \frac{3\pi}{2}\right]$, the only solution within this interval is $x = \frac{3\pi}{2}$.

**Analyzing the Second Factor: $\sin(3x)$**

The function $\sin(3x)$ is equal to zero at $x = \frac{\pi}{3}$, $x = \pi$, and $x = \frac{5\pi}{3}$. However, only $x = \pi$ falls within the given interval.

**Analyzing the Third Factor: $\sin(5x)$**

The function $\sin(5x)$ is equal to zero at $x = \frac{\pi}{5}$, $x = \frac{2\pi}{5}$, $x = \frac{3\pi}{5}$, $x = \frac{4\pi}{5}$, and $x = \pi$. Among these solutions, only $x = \pi$ is within the given interval.

**Combining the Solutions**

To satisfy the equation $\sin(x) \sin(3x) \sin(5x) = 0$, at least one of the factors must be equal to zero. From our analysis, we found that the only solution within the interval $\left[\frac{\pi}{2}, \frac{3\pi}{2}\right]$ is $x = \pi$, which makes the second and third factors equal to zero.

Therefore, the equation has only one solution in the interval $\left[\frac{\pi}{2}, \frac{3\pi}{2}\right]$, and that solution is $x = \pi$.

Thus, the number of solutions of $\sin(x) \sin(3x) \sin(5x) = 0$ in the interval $\left[\frac{\pi}{2}, \frac{3\pi}{2}\right]$ is 1.