Count rows where values in any of some columns are positive integers

Let’s say you have a some table consisting of four columns. The first one contains strings. The second, third and fourth may either be blank, contain a 0, or contain a positive integer. How do you count the rows where either the second, third or fourth column contain a positive integer?

I’m basically looking for a COUNTIFS where the different criteria are calculated using an OR instead of an AND.

Something like the formula below should do. It must be entered for array evaluation by Ctrl+Shif+Enter.


The well known trick with SUNPRODUCT would help to avoid the CSE:


I would clearly prefer to use a helper column doing the OR.