Question

The number of different signals which can be given from 6 flags of different colors taking one or more at a time, is
a)1958
b)1956
c)16
d)64

Answer 1

Solution

The number of different signals, which can be given from the first flag, is

$$N _ {1} = C _ {6} ^ {1} \cdot P _ {1}$$

Similarly the number of different signals, which can be given from the another flags, is

$$N _ {n} = C _ {6} ^ {n} \cdot P _ {n},$$

where $n$ is a number of flag.

So a sum of different signals is

$$\begin{array}{l} N = N _ {1} + \cdots N _ {6} = \sum_ {n = 1} ^ {6} C _ {6} ^ {n} \cdot P _ {n} = \frac {6 !}{5 ! \cdot 1 !} \cdot 1! + \frac {6 !}{4 ! \cdot 2 !} \cdot 2! + \frac {6 !}{3 ! \cdot 3 !} \cdot 3! + \frac {6 !}{2 ! \cdot 4 !} \cdot 4! + \frac {6 !}{1 ! \cdot 5 !} \cdot 5! + \\ + \frac {6 !}{0 ! \cdot 6 !} \cdot 6! = 6 + 6 \cdot 5 + 6 \cdot 5 \cdot 4 + 6 \cdot 5 \cdot 4 \cdot 3 + 6! + 6! = 1956 \\ \end{array}$$

Answer: b) 1956.