A certain model of remote-control Stanley garage door opener has nine binary (off/on) switches. The homeowner can set any code sequence. (a) How many separate codes can be programmed

Step-by-step explanation:

Given

------- on or off

Required

Determine the number of codes to be programmed

Since each switch has any of the on or off controls (i.e on or off), then:

The first switch can be coded in any of the 2 controls

The second switch can be coded in any of the 2 controls

The third switch can be coded in any of the 2 controls

Up till the ninth switch

So, we have that:

Hence, there are 512 possible codes

or

step-by-step explanation:

we need to solve for 'x' the following equation:  

we have that:

⇒

⇒ ⇒

factorizing, we get that:

  ⇒

then, the solutions are: or

nothing extraneous has been found!