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There is no sure answer. Anyone who claims to know so is using non-canon information.
Here's what we know in canon, however: - There were at least 10 "fleets" of Federation Alliance ships (includes UFP, Klingons, Romulans) - The 7th fleet had 114 ships at one point - When elements of the 2nd, 3rd, and 5th fleets attacked DS9 they had 627 ships - The final assault on Cardassia had a combined 1200 UFP, Klingon, and Romulan ships
So we really have no way of knowing, but an estimation of several thousand combat-worthy vesels seems to fit, plus thousands more support vessels.