How many different signals can be given using any number of flags from 5 flags of different colors?

Explanation

The signals can be made by using one or more flags at a time.
The total number of signals when r flags are used at a time from 5 flags is equal to the number of permutations of 5, taking r at a time, i.e., `.^(5)P_(r )`.
Since r can take the values 1,2,3,4,5, by the fundamental principle of addition, the total number of signals is
` .^(5)P_(1)+ .^(5)P_(2)+ .^(5)P_(3)+ .^(5)P_(4) + .^(5)P_(5)`
`=5+(5xx4)+(5xx4xx3)+(5xx4xx3xx2)+(5xx4xx3xx2xx1)`
=5+20+60+120+120
=325