Question

The no of different signals that can be given by using any no of flags of $4$ different colors is?

Answer 1

There are $4$ different flags given.

$\Rightarrow$ Signals using only $1$ flag $={}^4{C}_1=4$ ways

$\Rightarrow$ Signals using only $2$ flags $={}^4{C}_2=6$ ways

$\Rightarrow$ Signals using only $3$ flags $={}^4{C}_3=4$ ways

$\Rightarrow$ Signals using only all flags $={}^4{C}_4=1$ way

Total $=4+6+4+1=15$ ways.

Hence, the answer is $15.$