Does the grass seed I cut off while cutting the grass ever germinate?

In the early season, I cut the grass high to prevent dandelions. Well, I let it get a little long this time, so the grass was seeding.
I have wondered, if by cutting the grass, these newly formed grass seeds will produce new grass seedlings once delivered to the ground. I looked into the question at one time, and never got a definitive answer. Grass seed is not cheap, so if it means letting it grow real long and letting the seed dry on the stem in order to produce a germane seed, I will do it. I just may have to cut it two or three times in succession after the grass has reached 8 inches high. Unfortunately, this would likely produce more weed seed, as well.
I have two acres to cut, and really don’t care about most of it, just around the house.