Question

How many signals can be made by hoisting $4$ flags of different colours one above the other, when any number of them may be hoisted at once? How many with $5$ flags?

Answer 1

There are total $4$ Flags:-
Hoisted no. of flags Arrangements
$1$ ${}^4{P}_1=4$
$2$ ${}^4{P}_2=12$
$3$ ${}^4{P}_3=24$
$4$ ${}^4{P}_4=24$

Total no. of ways $=64$
Similarly with $5$ flags
Total no. of ways $={}^5{P}_1+{}^5{P}_2+{}^5{P}_3+{}^5{P}_4+{}^5{P}_5$
$=325$