The Fundamental Counting PrincipleColoured 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...

The Fundamental Counting Principle

  1. 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

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embizze | High School Teacher | (Level 1) Educator Emeritus

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(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

 

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