Direct your preferred web browser to united.com (see Resources.)
Enter the information for the flight you want to view in the "Book Flight" section on the left side of the page. If you are interested in the number of seats in only one of the cabins (Economy, Business or First), select that cabin in the drop-down menu. Otherwise, leave this option on Economy, the default. Click the "Search" button. This will open a page displaying all matching flights.
Select the flight you want, then click on the "View seats" tab beside the flight number for your flight. This will open a new window displaying all the occupied and available seats in the cabin you chose.
Count the number of available seats in this cabin on this flight.
To see what seats are available in the business cabin, click on the "Modify search" button on the left side of the main screen, then change the cabin selection to "Business." Click "Search." This will open another page with a list of flights. Find your selected flight again, and click "View seats" again. Here you can view available and occupied seats in the Business cabin. Count the available seats here. Also notice the row numbers; some planes in the Business cabin and the First cabin are identical.
Click on the "Modify search" button again and change the cabin from Business to First, then click "Search." Find your relevant flight again, and click "View seats" one last time to see the available and occupied seats in the First cabin. Check whether the row numbers match the row numbers for the Business cabin. If so, disregard the seats you see here. If not, count the available seats in the First cabin.
Add up the number of seats you counted in each cabin. This will provide you with the total number of available seats on the plane.