Garage door openers have 10 on-off switches on the opener in the garage and on the remote control used to open the door. The door opens when the remote control is pressed and all the switches on both the opener and the remote agree.

(a) How many different sequences of the on-off switches are possible on the door opener?

(b) If the switches on the remote control are set at random, what is the probability that it will open a given garage door?

(c) Remote controls and the openers in the garage for a given brand of door opener have the controls and openers matched in one of three sequences. If an owner does not change the sequence of switches on her or his remote control and opener after purchase, what is the probability that a different new remote control purchased from this company will open that owner’s door?

(d) What should the owner of a new garage door opener do to protect the contents of his or her garage (and home)?