Due to cross-chaining issues, I came up w/ a formula for the actual # of usable gears on a bike, which is less than the stated number by the manufacturer.

For instance, on my bike, which supposedly has 24 gears (3 in front, 8 in back), I only actually have 16 usable gears, and not 24.

The formula can be expressed in this form:

(# of speeds by manf) - (# of rear sprockets) = actual # of usable gears
ex. 24 - 8 = 16 usable

Note that this formula only works if you have 3 sprockets in the front, and any number in the back.

So my road bike with a 105 triple front DR, and 105 rear DR/Cog-set only has 20 usable? That's 30-10=20? That number seems low.

Let's start with the middle chainring. On that ring, I have full range of use of the rear cog set. So that's ten.

Next, look at the large chainring. On that ring I have the use of the smallest cog, all the way through the 3rd from the largest. After that, things start getting noisy. So we'll call that another eight.

And finally consider the small chainring. With the small one, I am able to use the largest cog, all the way to the 2nd from the smallest. That's nine.

Total: 10+8+9=27. Ok, the 2nd from the smallest on the small chainring may be impractical, even though it works ok on my bike. So we'll call it 26. Your formula was only inaccurate by 6, which is off by 20%. I'd say it's flawed.

There's usable gears and then there's gears that you use. Not necessarily the same thing.
Then there's gears that are duplicates that can still be useful, to avoid shifting back & forth on the big ring.
One can't really generalize this topic, because there are too many different combinations of rings and cassettes. Some choices are far more intelligent than others.

I saw a cheap Huffy kids bike, that was a 3x6.
24-36-42 with a 14-16-18-21-24-28 rear.
Do the ratio's on Sheldon's link.
Add a 12T cog and make it a 7 speed!
It's pretty much a lesson on what NOT to choose!

https://sheldonbrown.com/gears/

Just because the gears overlap does not mean you don't use them. Obviously your cadence is higher in a smaller chain ring than a bigger chain ring. Sometimes you want to emphasize your heart with a higher cadence or other times your muscles with a lower cadence. It is just being aware of your choices and how much to double shift when you switch chain rings.

On my triple when I am using the granny or biggest chain ring I usually don't use the two most extreme cassette gears. This is mainly to avoid the noise from cross chainining and also the wear that can be caused even though the wear factor really is not much of an issue with modern drive trains. I only use the middle six gears on the cassette when using the middle chain ring. So out of 30 gears I avoid using 8. If it were a 20-speed I would avoid using 4.

You've neglected to include in hyour post (a) the number of teeth on each chainring and cog and therefore (b) the number of repeated ratios on your remaining 26 gears.. and repeated ratios usually means in the range of 0 to 5 % difference. Likely that your 26 will come down to around 20 on that basis.

Overlap is good.

I once built a bike with a triple chainring and a 14/18 5-speed freewheel. My idea was to go through all 5 gears on one chainring, then shift into the next chainring and do it all over again.

It was a PITA to ride. It worked fine with nice small steps between gears until you had to make the chainring shift. Then, if you happened to be on a hill, you lost all of your momentum making the double shift and generally had to shift back to regain it. Overlap gives you the choice to make chainring shifts when it's most convenient.

Are you considering redundant gears?

As others have no doubt discovered, while riding around on fairly flat terrain I only use maybe 3 gears out of the many I have available. What I find myself liking more and more is my Gent's bike with it's wide ratio 3 speed Sturmey Archer AW hub made in 1962. Those three gears serve me well for all my leisure riding.

I guess another way to say it is that your gear selection depends on three factors. You select the chain ring based upon the gear inches needed, cadence depending upon whether you are emphasizing your aerobic capacity or muscular strength, and whether terrain and speed dictate whether are going to need higher top speed or acceleration to keep the pace you want or hang with the group you are in.

I got ripped off too!

They told me it was a 24-speed, but when I got it home, there was only 3 gears in the front and 8 in back. 3+8=11!