exoplanet - A star a black hole and planets around them

It's possible to have planets orbiting a binary pair of stars, your scenario of a close orbit, sometimes called "short orbit binaries". See here, also posted above in comments. In such a binary-system, nothing can orbit an individual star, but at some distance, plants can orbit and some systems like this have even been observed, listed in the link. The orbital dynamics is the same for a star-black hole short-orbit binary.

Now, there are problems. Black holes form out of very large stars and the formation is one of the biggest explosions in the universe, a Type II supernovas, and that's not very friendly to any planets in orbit. A star might survive it, planets would be harder, though it might be possible for new planets to form from nebula material remaining after the nova (I'm just guessing there).

A black hole could also form from a Neutron star accreting matter, but you still have the problem that the formation of a Neutron Star also only happens out of a Type-II supernova, so such a system has a difficult beginning.

Edit: While some planets have been observed around Neutron stars, these appear to be quite rare. 2 Neutron Stars have been observed with planets, out of over 1,600 Neutron stars observed. A type II nova is very planet unfriendly.

Theoretically a close gravitational capture is possible, but those are very rare, as stars rarely get that close. There's many stars that are known to orbit the super-massive black holes at the center of our galaxy (Andromeda galaxy too), but stars and stellar mass black holes are much more rare. A few have been observed, but they don't appear to be common. Such a system would be easy to observe, so the fact that there are only a few that have been noticed is evidence to them being rare. Here's a few mentions of them. One, Two, Three, Four.

From the 4th article, which is from 2011, so more may be known now, but it says:

Only about 20 binary stellar systems are known to contain a black
hole, out of an estimated population of around 5,000 in the Milky Way
Galaxy.

A 2nd problem is that a star feeding a black hole would create an accretion disk which would be very radioactive and not ideal for life on an orbiting planet. Maybe the planet could have a very thick atmosphere that might protect it, but that would also likely reduce sunlight reaching the surface. The star feeding the black hole would also be losing mass, and over time, grow smaller and provide less light and heat to the planet. Ideally, you'd want it to be a very slow feed. It's pretty far from an optimal life on planet situation.

So it is possible to have planets in this system not to be consumed by
this black hole but just follow their unique orbits

This part is certainly possible. Things can orbit a black hole at a safe distance without any problem. As for life, we don't know how common life is in other solar-systems so nobody can say how likely it might be, but it's theoretically possible, but, in my opinion, pretty far from ideal.