How many solar systems exist?

Only One. Solar refers to the sun derived from SOL. The "Solar System" is one of billions of STAR SYSTEMS though. There might be other start systems simular to the solar system but there still is only one solar system. The Solar System is a NAME of a star system albeit the one we live in but its still just a name. Names of a few start systems. Solar System (AKA Sol) Rigel Alpha Centauri Gamma Kruger etc.

Innumerable. A whole lot. At this point in our understanding, there are possibly an infinite number of solar systems. In order for something to be a solar system, there has to be at least one star and something orbiting it. It can be as simple and boring as a star with a cloud of dust around it or as complex as a multi-solar (more than one star orbiting around one another) with numerous planets, asteroid beltis, gas giant planets, and comets. A sun is just a star with something orbiting it. Our solar system (sometimes called "Sol") has one yellow star, and celestial bodies as follows: Mercury, venus, earth, mars , the kelper belt (asteroids), saturn, jupiter (both gas giants), uranus, neptune, pluto, xena (discovered in 2003), and 2 commets (haleys and bop). Not to mention all the moons. Within our galaxy, there are billons of stars, and we haven't discovered every solar system out there, not to mention the billions of other galaxies. Our galaxy is actually rather small in comparison with some that have been found. Some are friggin huge! It's higher than anyone could count. Literally more than there are grains of sand on our planet.

More than 1. According to NASA and other space "companies", they had found a lot of other solar systems, like ours. I'm not sure. Perhaps you can research about it.

Perhaps this person is actually too young to grasp the concept. A telescope fixed on Jupiter would show another planet and perhaps its 4 largest moons if you catch it at the right time. As for the other 250 or so planets so far known outside of our solar system, we know of those because we can monitor the light from stars and tell that sometimes the light is either closer to us or farther away at regular intervals. This supports the probability that a planet is orbiting and gravity pulls the star back and forth. Another method is seeing a change in the light coming from a star at regular intervals. If we measure that there's less light coming from a star every x number of hours, then again we can reason that a planet must be passing in front of it.

Innumerable. A whole lot. At this point in our understanding, there are possibly an infinite number of solar systems. In order for something to be a solar system, there has to be at least one star and something orbiting it. It can be as simple and boring as a star with a cloud of dust around it or as complex as a multi-solar (more than one star orbiting around one another) with numerous planets, asteroid beltis, gas giant planets, and comets. A sun is just a star with something orbiting it. Our solar system (sometimes called "Sol") has one yellow star, and celestial bodies as follows: Mercury, venus, earth, mars , the kelper belt (asteroids), saturn, jupiter (both gas giants), uranus, neptune, pluto, xena (discovered in 2003), and 2 commets (haleys and bop). Not to mention all the moons. Within our galaxy, there are billons of stars, and we haven't discovered every solar system out there, not to mention the billions of other galaxies. Our galaxy is actually rather small in comparison with some that have been found. Some are friggin huge! It's higher than anyone could count. Literally more than there are grains of sand on our planet.

More than 1. According to NASA and other space "companies", they had found a lot of other solar systems, like ours. I'm not sure. Perhaps you can research about it.