First of all, to make things simple, I round every measurement to the nearest 1/4 foot and round every product to the nearest foot. Doing so, I get 387'. Then I look at the layout and determine whether I think it will take more or less waste and factor in how the material being priced usually runs. Minimum waste factor is 5%. For yours I'd go a little higher then see how it comes out with full cartons.
387 + 6% = 410'
410' is 20.8 ctns. so ordering 21 ctns. would give me 414.12'. I think you should be able to cover 387' with 414' of material. The most waste is potentially going to come from the long rip cuts in the big room and hall. It wouldn't hurt to order an extra box. You might end up with an unopened box but it's nice to have some spares, just in case.
A few notes:
You made a typo on 2.3 x 5 = 20.7 but your total is correct. More importantly, 24 x 19.72 = 473.28, not 372.18.
Depending on how difficult it would be to get an extra carton should you need it, I'd order 21 or 22 cartons.

Ok thanks!

I've had time to look at this now, and I still don't see where you got 387'. And I guess I don't know what you meant by rounding every product to the nearest foot. By product, do you mean each box?

My numbers agree with JFC.

How I do it is round all of the inches in the room up to whole feet. So, for your living room, that 13'5" becomes 14', and the hall is 4'.

Then add in your 5% wastage. Then divide by the rounded down number of square feet in the box. which is 19 sf. I get 22 boxes.

That's odd. Everytime I add it, rounding to nearest 1/4, I get 397.

From wiki: In mathematics, a product is the result of multiplying.
13.25 x 19 = 252 (252 is the product)
2 x 5.5 = 11
5 x 11.25 = 56
2.25 x 5 = 11
3.25 x 17 = 55
3.25 x 1 = 3
Take the sum of all your products.
252+11+56+11+55+3=388
Multiply times a waste factor.
388 x 1.06 = 411.28
Divide the product by the amount in each carton.
411.28 / 19.72 = 20.86 ctns.
So 21 cartons, or if you choose, 22 cartons.
Depending on how you round your measurement numbers, you can swing the result a few square feet in either direction. Even if you get 397', 397 x 1.06 = 420.82, 420.82 / 19.72 = 21.34 ctns., so 22 cartons should be sufficient.
I do a lot of math in my head and always got dinged on tests for not showing my work even though I got the answers right. My math teacher would be proud!

BTW, I was using your written measurements and rounding them as I usually do.
If I go by your drawing, I come up with 397' also. Even if was being parsimonious and added only a 5% waste factor I come up with 21.14 cartons, so 22 should have you covered.

Also, if you are doing the installation yourself, paying attention to the lengths of boards can minimize waste.

1 - When you find a damaged board, trim out the damaged part and use it to start or end a row as soon as you can. Don't wait until the end of the project, because scavenging a pile of bad boards is not going to work well.

2 - Keep your off-cuts as short as possible. Lay out the boards roughly by length to select from and use ones that are just barely too long for the ends of rows.

This is a great post to make sure my installers order enough wood. Thank you!

Your math teacher would be proud. Mine would be ashamed. I did know what a product was, once upon a time. But I had manufacturing product in my tunnel vision.

We're not doing the installation, so hopefully the guys will only use 22 boxes. The tile people used a lot more than the salesman figured. They used a whole tile for every cut.