### Find the number of different signals that can be generated by arranging at least $2$ flags in order(one below the other) on a vertical staff, if $5$ different flags are available.

$320$

Step by Step Explanation:
1. A signal can consist of either $2$ flags, $3$ flags, $4$ flags or $5$ flags. Now, let us count the possible number of signals consisting of $2$ flags, $3$ flags, $4$ flags, and $5$ flags separately and then add the respective numbers.
2. There will be as many $2$ flag signals as there are ways of filling in $2$ vacant places in succession by the $5$ flags available.
By the fundamental principle of counting, the number of ways is $5 \times 4 = 20$
3. Similarly, the number of 3 flag signals is $5 \times 4 \times 3 = 60$, the number of $4$ flags signals is $120$ and the number of $5$ flags signals is $120$.
4. Therefore, the required no. of signals $= 20 + 60 + 120 + 120$$= 320.$