Four wires (red, green, blue and yellow) need to be attached to a circuit board. A robotic device will attach the wires. The wires can be attached in any order, and the production manager wishes to determnine which order would be the fastest for the robot to use. Required:
Use the multiplication rule of counting to determine four choices for the first wire, three for the second wire, two for the third and only one for the fourth.

24

Step-by-step explanation:

The topic here is COMBINATORICS.

The parent topic is PERMUTATIONS & COMBINATORICS.

Permutation deals with arrangement in a definite order while, as stated in the question here, definite order in not needed in Combinatorics.

Now, the multiplication rule of counting, also known as the rule of product, talks about the multiplication of the figures that represent the different ways of doing something.

For example, in this question, the robot needs to attach 4 wires to a circuit board. If you know how Physics or Electricity works, you'll that truly this is a combination matter and not permutation.

Putting/Connecting the 4 wires together (in a square shaped circuit for instance), the arrangement RGBY is different from RGYB or RBGY.

So there will be more ways to connect or combine these wires, than if we were to follow a definite rule like: "Red and Green must always stay together".

So using the multiplication rule of counting to determine 4 choices for the Red wire, 3 choices for the Green wire, 2 choices for the Blue wire, and 1 choice for the yellow wire, we have:

R4 x G3 x B2 x Y1  =  4 x 3 x 2 x 1  =  4!  =  24

The term "4!" means "Four Factorial".

the answer is ! ?