Given ^@7^@ flags of different colors, how many different signals can be generated, if a signal requires the use of ^@3^@ flags one below the other?


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Answer:

^@210^@

Step by Step Explanation:
  1. There will be as many signals as there are ways of filling in ^@3^@ vacant places in succession by the ^@7^@ flags of different colors.
    To find that we will use the fundamental principle of counting, which states,
    ^@“^@If an event can occur in ^@m^@ different ways, following which another event can occur in ^@n^@ different ways, then the total number of occurrence of the events in the given order is ^@m \times n^@.^@”^@
  2. The upper vacant place can be filled in ^@7^@ different ways by any one of the ^@7^@ flags.
    The second vacant place can be filled in ^@6^@ different ways by any one of the remaining 6 different flags and so on.
    Vacant Places Ways to fill
    		7
    		6
    		5
  3. Therefore, by the fundamental principle of counting, the number of different signals ^@= 7 \times 6 \times 5 = 210^@.
    Hence, the required number of different signals is ^@210^@.

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