### 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?

$210$

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 $210$. 