When a car has a dying battery - it can't hold a full charge and cannot deliver enough power - it is fairly obvious:
When the starter solenoid goes "CLICK-CLICK-CLICK" that is a battery without enough charge to turn over the engine. There is enough voltage to pull the starter solenoid in but once the starter motor tries to turn over the engine, the voltage drops and the battery can't hold the solenoid in anymore...so the solenoid lets go...which disconnects the starter motor so now there is again enough voltage to enough to pull the solenoid back in...and the cycle repeats with a "CLICK-CLICK-CLICK".
At this stage, the battery is close to death. Leaving the battery on a mains recharger overnight can perk it up somewhat, maybe get you another month or two, but you are delaying the inevitable. Time to replace your battery.
But there is another sound/response you may hear which isn't the battery, it is the starter. The giveaways are the solenoid pulls in no problem, the starter will crank the engine but very slowly, and then maybe starts cranking a bit faster. Or won't crank or stops cranking after a moment - with the solenoid still in. The clue here is the solenoid never lets go. Because the solenoid is still working fine, but the windings in the starter motor, or some rotor segements have failed.
(Failed rotor segments are old school notorious, when just one out of the 10 to 20 segments isn't working, and the starter lined up exactly on that one. And hitting the starter with a hammer would fix it, until one day you've hit it too many times and it doesn't work at all...)
A few years ago my Falcon failed to start, and I braced myself to change the starter. But there was no click of the solenoid, just the dash lights went out in the start position. It turned out that the starter solenoid electrical connection had come loose. Phew, dodged that one - just reconnect the connection. The clue here: no solenoid click at all.
Another time it failed to start, that time no click at all, no dash light going out. It turned out to be the auto transmission start-in-gear interlock.
But then, after 13 years of trouble free starting it was again proving difficult. I couldn't quite nail it down - was it the battery? Was it the starter? Some days I thought it was the battery, some days the starter. So, I did nothing for a few months. Until a very hot day when it cranked for a bit, and then stopped, never to try turning over again. The clue - the battery was pulling the solenoid in just fine, but the starter motor was doing nothing. Argh. And it's a very hot day. By the way, you can't bump start an automatic Falcon.
Suffice to say a mate of mine towed me home, and first thing the next day I went and sourced a new starter. My mate said to head over to Ashdown-Ingram in Wangara WA, they would look after me. He was correct. These days starters are throw away, no longer exchange, RRP on an after market starter was AUD$170(2019) but they gave it to me wholesale $92. Thanks! Note that some models of Falcon have 9 teeth on the starter, some have 10. Mark one tooth, and then count, don't get it wrong.
Removing the old and installing the new - it is possible to do without say taking the engine out, but it is hard. You jack up the passenger side of the car, and go under. There are two bolts holding it in. One at the bottom which is easy enough to get to. One at the top which is not easy to get to, and you can't see it. You have to go by feel. I had a socket on a 20cm extension and I just kept poking around in the general area until it felt like I was on the bolt. Then I plugged in the ratchet, carefully, and undid/redid. You will get frustrated. You will lose skin. But you can do it. Years ago I put a little mirror on a bendy piece of wire, I used this to see behind the starter to get my bearings, and tried again. Stay calm, be patient, you'll make it.
Once you have your new starter installed you won't believe how quickly the engine turns over! You don't realise how slow it becomes until you compare it to a new one!