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You can do this by using the count() function in your field "Count".
1. Double click on the Count field (I suppose this is a text box).
2. Click on the Default Value (fx) button in the Data tab.
3. Click Insert Function, and select count function from the list.
4. Click the link to insert a field. Select your first check box, say Company 1.
5. Click Filter Data. Add a filter to set Company 1 equal to TRUE (to count only if the check box is selected), then click OK 3 times.
6. Next to the first Count() formula, enter a plus + then repeat steps 3-5 to the rest of the check boxes. Your formula should look something like this:
count(field1[. = string(true())]) + count(field2[. = string(true())]) + count(field3[. = string(true())]) + count(field4[. = string(true())])
7. Click OK when done. Then make sure to check the check box "Update this value when the result of the formula is recalculated" just below the Default Value field in Data tab.
8. Click OK.
Try selecting/deselecting from the check boxes and see if the count works!
Thanks,