The Fundamental Counting Principle
- Coloured flags are often used by ships to signal at sea. Assume that a ship can hoist up to five different flags and that changing the order of the flags on the mast results in a different signal. How many different signals are possible for each arrangement?
a) If all five flags are used
b) If four flags are used at a time
c) If at least one flag is used
(a) If 5 flags are used then there are 5 choices for the first flag, 4 for the second, 3 for the third, 2 for the fourth and 1 choice for the fifth flag.
The number of different signals using 5 different flags is 5!=120
(b) For any group of 4 flags there are 4!=24 different possible signals. There are 5 different groups of 4. (If the flags are numbered 1-5, then there is a group excluding 1, excluding 2, etc...)
Or as above, there are 5 choices for the first flag, 4 for the 2nd, 3 for the 3rd, and 2 for the 4th.
The number of different signals using 4 different flags is 120
** This is the same as using 5 flags -- once you have chosen the first 4, the fifth flag is the only available flag left. Note that the presence ar absence of the fifth flag could itself be a signal so 1-2-3-4 is different from 1-2-3-4-5 **
(c) If you use at least 1 flag:
There are 5*1!=5 different signals using 1 flag. (5 choices of 1 flag and 1 way to arrange it.)
There are 5*4=20 different signals using 2 flags. (5 for the first flag and 4 for the second.)
There are 5*4*3=60 different signals using 3 flags.
As above there are 120 signals using 4 flags, and 120 signals using 5 flags.
The total number of different signals using at least 1 flag is 5+20+60+120+120=325