Producers used psychology tricks during casting.
When the producers would see the child actors at auditions they would employ a number of casting tricks. One trick included putting a bunch of distracting children-toys on the table. Kids who paid attention to the toys were passed over immediately!
Mrs. Brady had two actors.
Florence Henderson wasn’t the only Mrs. Brady that the show had seen! Initially Joyce Bulifant, a more comedic and zany actress, was brought on board. She was used extensively in screen tests and as an aid during casting in the rest of the show. However, the producers decided to go for a more subdued Mrs. Brady and thus brought on Mrs. Henderson.