Question

The flag of a newly formed forum is in the form \[\square \square \square \] of three blocks, each to be colored differently. If there are six different colors on the whole to choose from, how many such designs are possible?

Answer 1

Hint: The first block will have 6 colors to choose from and hence it will have 6 ways. And the second block will have remaining 5 colors to choose from and hence it will have 5 ways. Again the third block will have 4 colors to choose from and hence it will have 4 ways.

Complete step-by-step answer:
Clearly, the first block can have any of the 6 colors to choose from and hence there are 6 ways in which the first block can have a color.
Now the second block can have any of the remaining 5 colors to choose from and hence there are 5 ways in which the second block can have a color.
Now the third block can have any of the remaining 4 colors to choose from and hence there are 4 ways in which the third block can have a color.
Hence, by the fundamental principle of multiplication,
The required number of ways \[=6\times 5\times 4=120\].
Hence there are 120 designs possible.

Note: Remembering the concept of fundamental principle of multiplication is the key here. The fundamental counting principle (also called the multiplication rule) is a way to figure out the number of outcomes in a probability problem. Basically, we multiply the events together to get the total number of outcomes.