Question

How many different signals can be made by hoisting $6$ differently coloured flags one above the other, when any number of them may be hoisted at once?

Answer 1

No of ways for hoisting $6$ different flags out of $6$ given flags = ${}^6{P}_6=720\text{ways.}$

No of ways for hoisting $5$ different flags out of $6$ given flags = ${}^6{P}_5=720\text{ways.}$

No of ways for hoisting $4$ different flags out of $6$ given flags = ${}^6{P}_4=360\text{ways.}$

No of ways for hoisting $3$ different flags out of $6$ given flags = ${}^6{P}_3=120\text{ways.}$

No of ways for hoisting $2$ different flags out of $6$ given flags = ${}^6{P}_2=30\text{ways.}$

No of ways for hoisting $1$ different flags out of $6$ given flags = ${}^6{P}_1=6\text{ways.}$

Hence, total number of signals $=720+720+360+120+30+6=1956\text{ways.}$