How many different signals can be given using any number of flags from 4 flags of different colours?

Explanation

These signals can be made by using 1 or 2 or 3 or 4 flags at a time. Number of signals made by 4 flags:
(i) taking 1 at a time`=""^(4)P_(1)=4,`
(ii) taking 2 at a time `=""^(4)P_(2)=(4xx3)=12,`
(iii) taking 3 at a time`=""^(4)P_(3)=(4xx3xx2)=24,`
(iv) taking 4 at a time`=""^(4)P_(4)=(4xx3xx2xx1)=24.`
Hence, by the fundamental principle of addition, the total number of signals `=(4+12+24+24)=64.`